The 7DII sounds as if it will be a great sports/action camera... after the "new model" dust settles a little, I will be getting two of them to replace my pair of five year old 7D, which have been excellent cameras but now have over 100,000 clicks apiece on them. I use them mostly for sports/action (and have a 5DII full frame camera for more sedate subjects).
That said, based upon your description I think 7DII might be overkill for your purposes. Your 60D... or perhaps an upgrade to a 70D... sounds perfect for your needs and is quite capable for action/sports shooting.
I would highly recommend putting your money into lenses, instead.
First, keep your EF-S 17-55/2.8 IS USM. It's an excellent lens.
Now, get a wide angle for landscape shots... I would suggest the Canon EF-S 10-22mm USM as one of the best in its category. (the entry level 10-18mm STM IS is a bargain at about half the cost, but more lightly built, has less range and isn't quite as sharp from edge-to-edge. Personally I won't be trading in my 10-22mm!). A 10-22mm costs $600, plus another $25 or so for the separately sold lens hood. (There's a $50 rebate on the lens, right now.)
Next, trade up from your two longer lenses for something better. The EF-S 18-200mm is okay, but doesn't have USM focus, so will not be particularly good for sports/action.
There have been several 70-300 and 75-300 lenses... not sure which one you have. Hopefully not the 75-300 that's often sold in kits with cameras (or separately for $200 or less) and is one of Canon's least successful lenses. The more expensive 70-300 IS USM is a pretty decent lens (while the premium 70-300L and more compact 70-300 DO are truly excellent... but far, far more expensive at $1450 and $1400 respectively).
I'd recommend a 70-200mm. There are several good ones to choose among. Canon offers two with IS, which I'd highly recommend. The 70-200/4L IS USM is lighter, more compact and more affordable (tho not "cheap" by any means). This lens doesn't come with a tripod mounting ring... it's sold separately (Canon's costs about $150, but there are 3rd party clones available for around $50 that seem fine). The f4 version of the lens will set you back about $1300 typically (less a $150 rebate right now).
Alternative is the 70-200/2.8L IS USM Mark II... bigger, heavier, comes with a tripod ring and is the latest and greatest of the line-up... but it's expensive: about $2200 ($100 rebate until October 25). If you could find a good one lightly used, the original 70-200/2.8L IS USM is also a very good lens selling for around $1500, more or less depending upon condition.
The reason I recommend these lenses is that they are tops for sports photography... or just about anything else. The two models I recommend both use fluorite elements and are capable of making images sharp enough to make your eyes bleed. USM focus drive is far faster acquiring and better tracking moving subjects. Combined with your 60D... or perhaps a 70D... you will have little trouble keeping up with most sports situations.
If you find you miss the 300mm you enjoy now, both these 70-200s can be used with 1.4X teleconverters. The Canon EF 1.4X III is rather pricey at about $450, but the 1.4X II is almost as good in nearly all respects and can be found lightly used for around $250 typically. There is not enough difference between them that I've bothered to upgrade from my II versions of 1.4X and 2X. I haven't used them, but purportedly the Kenko Pro 300 1.4X DG and DGX teleconverters rival the Canon for quality (and currently sell for $250 new). The $150 Kenko MC-4 1.4X DGX is also quite good for the money... may even be sharper in the center of the image, though not as sharp at the edges and corners.
Personally I use a Canon EF 300/4 IS USM a lot, when I need a longer lens. It is very portable and hand-holdable, plus works quite well with a quality 1.4X teleconverter if needed. I bought mine use (practically unused, actually) for $900. Normally it sells for $1450 (there's a $100 rebate, right now).
Alternatively, the Canon EF 100-400L IS USM that costs around $1700 (currently a $200 rebate) is a popular sports/action zoom giving up to double the reach of a 70-200. At the long end, it's an f5.6 lens. That's fine in reasonable light, but may be a problem in low light conditions. It's also slightly bigger than a 70-200/2.8. It's a push/pull zoom design... unusual among today's lenses. Folks either really love or really, really hate this design of zoom, it seems. Hard to say without trying it. I didn't like them in the past because I found it hard to get a steady shot with a push/pull zoom... but to be fair, this lens has IS which should help a lot.
I use a few third party lenses, but mostly stick with Canon OEM simply because they generally are the best in their class and are very full featured. Plus, Canon lenses are more likely to be both backward compatible with my old film Canons and future-proof for Canon DSLRs I might buy in the future. I tend to only look at 3rd party lenses when they offer something that just doesn't exist in the Canon line-up. But those are pretty few and far between, since Canon has the largest system of lenses of any manufacturer.
Yes, a full frame camera might be nice for landscapes, thanks to their potential for great detail in images and enlargeability of their image files (I use a 5D Mark II, personally)... Also, they are better able to handler low light situations, thanks to lower image noise (which you didn't mention as a concern.... besides you should be able to use 60D to ISO 1600 easily, or 3200 and even 6400 with a little more care and work).
Most of the time and with reasonably good technique, 18 and 20MP crop sensor cameras can produce excellent landscapes printable up to around 16x20 or 18x24". However, if you have occasional need for extreme detail and enlargeability, there's a simple trick you can do with a crop sensor camera. Turn it to the vertical/portrait orientation and take three or more shots, then combine them later using a panorama software. In fact, there's a process called Gigapan that can use virtually any camera to make as many as 500 shots and later combine them into one massive image (often more than a gigabyte in size), that has almost unlimited detail. This is done using a computerized, robotic tripod head that insures all the images are neatly aligned. The whole process is based upon NASA exploratory photo techniques. Pretty cool... more info at:
http://gigapan.com/Anyway, if I were you I wouldn't necessarily spend the $1800 for a 7DII. Either keep the camera you've got, or upgrade to a 70D instead. Then put the money saved into an EF-S 10-22mm USM and the EF 70-200mm IS USM of your choice (f4 or f2.8), selling off the 18-200mm and 70-300mm lenses you currently have. If you wish, add a 1.4X teleconverter.
If you don't already have them, also get lens hoods for all your lenses (L-series such as the 70-200s include the hood). A properly fitted lens hood is one of the least expensive ways to both improve your images and give some reasonable protection to your lenses against accidental bumps.
Also, if you don't already have one, get a very high quality 77mm circular polarizing filter, such as the B+W Kaƫsemann ($135) or Hoya HD2 ($185). This will be a very useful filter for landscape shots in particular, and will fit both the 10-22mm and 17-55mm lenses. (It also will fit 100-400, EF 300/4 and 70-200/2.8 lenses. But not 70-200/4 that uses a 67mm filter instead. But, frankly, it's generally less necessary to use C-Pol on a telephoto lens, than on a wide to standard/short tele.)
Finally, for landscape photography a good, stable tripod is highly recommended. Yes, it allows you to use lower shutter speeds and all... but one of its biggest benefits is that a tripod makes you slow down and think about the shot you're about to take, often making for better results.