I would eliminate the TX10.
I have one, as one of the EDC tool in my everyday bag, stored inside a semi-hard case for further protection on top of it is a tough camera (waterproof, low temperature proof and shock proof).
Unluckily before the end of its 12 months warranty period the case developed a hair thin crack line without dropping or bumping into any thing. Service centre replaced the case for free. 2 months later a crack line was developed again without reason. After a tough war the case was replaced again for free but service centre claimed the waterproof ability should be intact despite of a cracked case. Few months later, a cracke line was found again! I gave up on it.
When i bought my used TX10 it had a image stabilization error, but it was a well used camera and for $20 i couldn't complain, the camera still worked fine.
My favorite model is the TX5, it's slightly thicker but the images come out looking slightly better for some reason, also less slippery and more affordable to get today. They have one issue, the sliding panel on the front of the camera is polished and painted over, which means the paint tends to flake away with use, but some undamaged TX5s still exist
Regardless of the model, a brand new replacement battery will be needed.
Also worth adding is that the older T series cameras are OK too, but i wouldn't spend any significant amount of money on them.
I'il also take the opportunity to mention that if i compare the images i get with my Experia 1/5 series phone with a T/TX series camera, the phone comes out as the winner, and the phones themselves are almost as tiny as the cameras.
So, from a practical perspective and also money-wise, today it makes more sense to order a refurbished 2019 Experia 1 from Aliexpress and use it strictly as a camera(they're also waterproof and have better screens) than it is to spend the same amount of money on an older T/TX series camera, which were never designed for good IQ to begin with, they were Sony's fashion series cameras.
I only mention this because i recently bought an Experia 1 J9110 for very little money, while cybershot prices have exploded for some models, all of a sudden they are cult cameras or something. The image quality of these 1/5 Series Experia phones took me by surprise, they look just like what your average pocket camera would produce and i'm very picky about my pocket cameras. The phones also have a dedicated shutter button and things like color spectrum sensor(better AWB) and they work very well overall.
We live in interesting times and as a person who enjoyes testing out affordable older cameras, i can only recommend products that make sense to me. When i used to buy used cybershot cameras, i'd only spend a few bucks on most of them, now the same cameras sell for $150-$200 on ebay and i find that insulting to my intelligence, so i'm looking for alternatives.

Many cybershot cameras would struggle with this image
