Bridgeport is a poor world for testing because it's difficult to work around inactive sims all across town getting stuck in elevators and instantiating inside of LN apartment NPC-only units unable to get out thus causing routing issues. Twinbrook though is known to be gentle to play as would be Sunset Valley and Riverview. From what you have described, yes I think this card is aging out on you and the Pets/Seasons combo is just too much for it now, as are some other GPU intensive games.
On the more current range of Nvidia cards, for a desktop a GTX 1050/1050Ti or the AMD equivalents should be fine. A 1060 would be stronger, any higher than that and I believe any in the newest 2000 series would provide more strength than TS3 would ever need as the program can only utilize 800 MB of vram. But other games could benefit from going higher. Note though that the stronger cards, any of these, require more power to run and generate more heat so one has to be sure that the computer's PSU and cooling systems can handle them. Your CPU also needs to be strong enough to avoid bottlenecking, what has it got for a processor?
