The desktop and normal applications are easy for it to render, when running a game it utilizes the RAM and GPU at a much higher level. I'd contact AMD and ask them about it, then if you still see the problem the next step contact the manufacturer and RMA it. It could be something with the RAM or GPU chip itself that is not working as designed.
One other thing to look at is the settings in Adrenalin and the game to make sure you have AA and AF set like you want it. Also is it in every game? The 6600 XT has 2 GB less RAM than the 2060, but that would only show up as being slower or the game may limit what settings you can use if it detects the RAM amount, and even then it should still render correctly, just slower.
If the textures are noticeably worse on the 6600 XT I'd say the card has some sort of hardware problem, although it's still tough to say sometimes, lots of variables.
Good luck!