Being a Sony shooter since 2008, I have used a wide array of Sony bodies, including a NEX 6. I would keep the lenses and upgrade the body. The A6600 is way ahead of the NEX 6 in terms of performance. You can shoot at a higher ISO and not worry. The IBIS is worth the price of admission alone. You can also use the panorama feature to compensate for not having an ultra wide angle lens. Clear Image Zoom will give you all the reach you would need. Granted, these last 2 features are for JPEG only, but the trade off is worth it. The JPEG files are highly manageable if you desire to use them. There is always RAW for the rest if that's your usual setting.

I have a Sony 18-135 on my A58. It is my do-everything travel lens. For me, it is a very good lens. If I need more focal length (seldom when traveling), I have the 2x CIZ (Clear Image Zoom) built into the camera.
I also have a Sigma 10-20mm, which I like alot, but I don't carry it when travelling. If I need wider view, as noted above, I use the internal panorama function, which works very well.
I also have a Tamron 28-300, but it is heavier and I seldom use its reach when travelling. (I do like using in a zoo.)
I just prefer not carrying multiple lenses, and changing them, when travelling.
I do often carry a +1 or +2 close-up "filter" as well as a polarizer, both of which, for me, occassionally come in handy.
In the film days, I travelled to Japan and Panama with a nifty-fifty and a 70-210 zoom, which covered everything I needed (from poison frogs to Mt. Fuji). The 18-135 on crop-frame gives me more range in one lens, with equal or better IQ (from observation, not testing).
I highly recommend it. I bought mine used at KEH. Hope this helps.

I agree. Conventional wisdom says upgrade the glass before the body. But in this case the NEX line has been so substantially eclipsed by the A6xxx models (even by the now-venerable A6000 let alone its entry-level replacement the A6100) in image processing, autofocus, ISO etc. that I'd go for the new body.

But I'd consider the just-released and best-selling A6700 in the mix too. Not much more money than the A6600 and much closer to state-of-the-art in 2023.

In fact they are the same price at Amazon and elsewhere, but you may well find the older model discounted by some vendors.

https://www.amazon.com/Sony-Alpha-6700-Interchangeable-Recognition/dp/B0CBN8Y77Z/ref=sr_1_1?crid=1D51TRIX2FYJ&keywords=sony%2Ba6700%2Bbody&qid=1696138759&sprefix=sony%2Ba6700%2Bbody%2Caps%2C74&sr=8-1&ufe=app_do%3Aamzn1.fos.17f26c18-b61b-4ce9-8a28-de351f41cffb&th=1

https://www.amazon.com/Sony-Alpha-A6600-Mirrorless-Camera/dp/B07X43B6FD/ref=sr_1_2?crid=3LWPY7RDEMYJJ&keywords=sony%2Ba6600%2Bbody&qid=1696138701&sprefix=sony%2Ba6600%2Bbody%2Caps%2C75&sr=8-2&ufe=app_do%3Aamzn1.fos.765d4786-5719-48b9-b588-eab9385652d5&th=1

Why don't you take your camera and 16-55mm lens outside. Take a few shots at 16mm, then zoom to 18mm and take those same shots again. Now come inside and repeat the same experiment again. Look carefully and see if that 2mm or so represents a critical difference to you. And yes...I know...you will have to estimate where 18mm is on the zoom ring.

I don't have any Sony cameras. But I shoot with both full frame and crop cameras. Each of them has a identical 24-120mm zoom most of the time. I've been shocked at how rarely I've encountered any inconvenience at the wide end.