"All-in-one" lenses may be convenient... but they ALWAYS have to compromise in some ways.... as you have learned with your Tamron 16-300mm.
And now you propose buying another "all-in-one"? The 16-300mm you have now is an 18.75X zoom... very extreme. The 18-400mm is even more so. In fact, it's the most extreme anyone has made, to date... more than 22X! That new Tamron is surprisingly good for such an extreme zoom.... but if you are looking for significantly improved image quality, good enough for prints to frame and hang on the wall, look elsewhere.
Would the Canon EF-S 13-135mm be better? Yes, but how much depends upon which one. There are three different versions of the EF-S 18-135mm. The latest two are identical optically and have better image quality than the earliest version: the EF-S 18-135mm IS. Canon revised the optical formula and added faster, quieter autofocus with a stepper motor in the 2nd version, the EF-S 18-135mm IS "STM". They left the optics alone, but introduced their even faster (2X to 4x, they claim) "Nanon USM" focus drive motor in the 3rd version. This last version also is currently the only Canon lens designed to be usable with the accessory PZ-E1 Power Zoom module, which was introduced along with the lens and might be of interest to people shooting videos. The 1st (not STM or USM) is rarely seen new any longer, sells used for under $200. The 2nd "STM" version sells new for about $400 and used for $225-$250. The 3rd "USM" version sells new for $600. It is new enough that there aren't a lot selling used yet. It's not in stock right now, but Canon USA occasionally has refurbished EF-S 18-135mm USM for sale for around $475.
The EF-S 18-135mm STM or USM would be a nice "general purpose, walk-around" lens, covering the range from moderately wide to moderate telephoto. The STM or USM is about the same size and weight as your Tamron 16-300mm. The Tamron 18-400mm is a bit larger and about 6 oz. heavier than either of them.
You can compare the image quality of the EF-S 18-135mm STM or USM with your 16-300 or the 18-400mm here:
https://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=1045&Camera=963&Sample=0&FLI=5&API=1&LensComp=953&CameraComp=736&SampleComp=0&FLIComp=4&APIComp=0To me it's clear the 18-135mm lens is superior... especially compared to the 16-300 you're using now. Those are magnified test shots (see
https://www.the-digital-picture.com/Help/ISO-12233.aspx for a full view of the standardized lens test target being used). You can compare similar focal lengths and how the lenses perform stopped down. In some cases, you can see how they perform on different cameras (all crop sensor APS-C models, with any EF-S lens or those Tamron "Di II" models). You can also change the lens selection to see how others compare against each other.
Your original post suggests you don't use the longer focal lengths much, but you already have alternatives for that purpose... specifically:
1. EF 70-200mm f/2.8.... I'm not sure which version you have. Canon has produced four different ones. Still in production, the oldest doesn't have IS, is the least expensive and optically the weakest of the bunch. The 70-200/2.8 with IS was improved optically and had image stabilization added. Even better was the II version of that, with fluorite added. It's still in stores new. A III version with IS has recently been introduced, but I don't know how it compares to the II version.)
2. EF 70-300mm.... Once again, I'm not sure which version you have. Canon has produce four different ones. Currently they offer the EF 70-300mm "II" IS USM, a good performer with "Nano USM" and an LCD distance scale. The model that preceded it has USM (not as quiet, but just as fast) and IS, with similar, very good image quality. There also is currently a premium "L" version, which is larger and heavier, better sealed for weather resistance and is the only one of the bunch that can optionally be fitted with a tripod mounting ring. Finally, a more compact (but not lighter weight) "DO" or "diffractive optics" version was the most expensive of the Canon 70-300s, discontinued a couple years ago, with pretty decent image quality, IS and fast USM focus drive.
3. EF-S 55-250mm... There have been three versions. All three have IS and there isn't a lot of difference between the first two, both of which use slower, noisier micro motor focus drive. The latest "STM" version has faster, quieter AF. I believe it may be closer focusing, too. All of them are optically superior to the Tamron 16-300mm you're using.
You also have a compact, short/fast (large aperture) telephoto...
4. EF 50mm f/1.8... Again, there have been three versions. All of them are designed as low cost, entry-level lenses. Acting as short telephotos on an APS-C camera like 70D, they are popular for portraiture, as well as low light conditions. The original was discontinued many years ago and replaced with the "II". They both use slow, noisy micro motor autofocus. The "II" was a cheapened version, with plastic bayonet mount instead of metal. It also doesn't have a distance scale, the way the original did. The II commonly sold for about $125 new, less than $100 used. The original became popular for it's better build and prices for used ones were often around $200... more than it sold for new. The current model has revised and improved optics, the best image quality of the three, plus it uses faster/quieter "STM" focus drive.... and it's got a metal bayonet mount, even though it sells new for the same price as the earlier version: $125. A nice thing about the II and the STM is that they don't really need a lens hood... their front elements are recessed a bit, pretty well shaded and protected from bumps or stray fingers. The STM can safely be manually focused at any time, though it has a pretty skimpy focusing ring to do so. (The micro motor "II" should not be manually focused without first turning AF off at the switch on the lens. Failing to do so an manually overriding AF will damage the focus mechanism of lenses with this type of focusing drive motor.)
Do you REALLY need a full frame camera "eventually"? Unless you are planning to make big prints... really big prints... I'd argue, probably not. A full frame camera requires full frame lenses, which are necessarily bigger, heavier and more expensive. FF lenses need to use larger lens elements to be able to produce large enough image circle to cover the larger sensor. I would HIGHLY recommend you first try an updated APS-C camera instead, such as the 80D successor to your 70D. The 80D has a lot of very nice upgrades over the earlier camera... a new, higher resolution sensor, better AF system and much more. In time there will be yet another successor (90D?), but who knows what features it will have. In the meantime, the 70D is quite capable, with a lot of good features (the 7D Mark II still uses the 20MP sensor found in the 70D).
Your problem is not the camera. It's your lenses. You've got a number of inexpensive, entry-level quality lenses, a whole lot of duplication of focal lengths and are being just plain lazy trying to "make do" with an all-in-one zoom that compromises in a lot of ways, rather than carry and - when needed - swapping between a few less extreme, much more capable lenses. There are lenses that would be better choices.... either offering better image quality or expanding your kit's capabilities... or both. But there are also quite usable lenses already in your kit... if you would just use them.
Some good news is that you have some experience with "better lenses", using your 70-200mm f/2.8L. A 3X (200mm divided by 70mm) zoom like that is much better than an 18X+ all-in-one zoom, as you've clearly discovered. The problem with that particular lens is that a 70-200mm f/2.8 is rather large an heavy. But there are alternatives. For example, the EF 70-200mm
f/4 lenses are about 2/3 the size and weight of the f/2.8 lenses. (They also cost less and all of them use fluorite, which helps makes them sharper than the first two Canon 70-200/2.8Ls, which don't use fluorite.)
Upgrading that 70-200 wouldn't help with duplication, though. (You've got FOUR lenses covering similar telephoto range: 16-300, 55-250, 70-200 and 70-300mm!).
Also, don't be an "L-coholic"... Don't get hung up on the Canon L-series designation. Yes, most L-series are quite good. But some of them are not all that great. PLUS, there are a lot of excellent Canon lenses that AREN'T L-series for one reason or another. For example, no EF-S "crop only" lens is an L-series, regardless how good it is. By definition, an L-series must be "compatible with all EOS cameras" and the crop lenses cannot be used on any full frame or film cameras. In fact, some EF-S lenses rival or out-perform L-series. For example, someone shooting with an APS-C Canon would be much better served buying an EF-S 17-55mm f/2.8 IS USM lens than an EF 17-40mm f/4"L" USM. The EF-S lens has a wider range of focal lengths, a stop faster max aperture, image stabilization which the L-series lacks... In fact, the 17-55 even has slightly better image quality than that particular L-series (both are quite good... but the 17-55 is a wee bit better at various aperture and focal length settings it shares with the 17-40L).
Rather than more duplication, you might do better to look for lenses that expand your options... Lenses that augment what you have now. Just for example, if you like to shoot landscape/cityscape/scenics, a wider lens might be nice. The widest you've got now it 16mm, which is okay, but it's only possible with the relatively compromised all-in-one zoom. The widest other lens you've got is a fairly modest 18mm with the kit lens (18-55mm). There are a number of very good ultrawides. But, again just for example, Canon themselves offer two: EF-S 10-18mm IS STM (under $300) and the EF-S 10-22mm USM (around $600 new). Both have excellent image quality. The more expensive lens is better built with larger apertures (f/3.5-4.5) and faster, higher performance USM auto focus. The cheaper lens is more plasticky, but it's also smaller, lighter and it has image stabilization (not necessary on an ultrawide, but nice on any lens).
Or, maybe your "thing" is shooting macro and close-ups. You don't have a lens to serve that purpose... and actually may not need one. You could just get a set of macro extension tubes. The Kenko set for Canon is high quality at a reasonable price, around $130 new. There are also Vello, Opteka and others that cost around $75... more plasticky and lighter built, but probably fine for occasional use with lighter weight lenses. Macro extension tubes are used in between a standard lens and the camera, to make the lens focus closer and render higher magnification. They literally can be used with any lens that can fit the camera. The type of tubes mentioned above have electronic contacts that allow autofocus to continue to work and let you adjust the aperture. (There are much cheaper tubes without electronic contacts... often under $25 or even under $15... but you should avoid those for use with modern electronically controlled lenses. The main issue is that it's near impossible to adjust the lens aperture. AF also won't work, but macro can be done with manual focusing.)
Of course, there are also true macro lenses, capable of focusing quite close and rendering high magnification. I generally recommend a lens around 90, 100, 105mm for general purpose use in the field. But there are more compact options too, including Canon's own, relatively affordable EF-S 60mm f/2.8 USM. Around the highest possible magnification, a shorter focal length like this doesn't give you as much working room between your subject and the front of the lens... but if you aren't doing a lot of full 1:1 magnification shooting, a lens like this can work fine. It also could replace your 50mm f/1.8 as a portrait lens. The only problem is f/2.8 instead of f/1.8. But a possible solution for that is the Tamron 60mm f/2 macro lens... very close to f/1.8, only one-third stop difference. The Tamron 60mm is considerably more expensive than the Canon 60mm, though. The Tamron also has slower micro motor focus drive, but no macro lens is particularly fast focusing. Macro lenses are slower focusing by design. They emphasize precision over speed.
Those are just a few examples that may or may not be what you're looking for. However, I want to point out that all the options mentioned are "crop only" lenses that will work great on your 70D, but won't be usable on some mythical, future full frame camera. One of the advantages of a crop camera like 70D is that it can use both crop and full frame lenses equally well. A full frame camera, on the other hand, can only use full frame lenses (which - once again - are bigger, heavier and more expensive).
Yes, I think the EF-S 18-135mm IS STM or USM would be a good idea... A "general purpose, walk-around" lens with very good image quality and AF performance. it would replace your 16-300mm and 18-55mm lenses. If and when you need a more powerful telephoto, switch to either your EF-S 55-250mm (if it's the later STM model) or the 70-300mm (assuming it's the IS USM or IS USM "II"). Both of those less extreme zooms will give you MUCH better image quality than you see now from the 16-300mm.
Anything else you carry would be optional and is entirely up to you. If you prefer to haul around the 70-200mm f/2.8 instead, great. Or maybe you'll choose to add an ultrawide or a macro lens or to carry your 50mm f/1.8 too, for low light conditions or portraiture.
Hope this helps!