Processor may be a limiting factor going forward, as well as PCIE Gen 3 slot on motherboard, but you should still be able to get the majority of performance put of most GPUs. Look for Radeon 6600 or 6600 XT in your local market, or if you can find an RTX 2070 for a good price.
Going newer, a 4060 would also be a good choice, and DLS3 would be a good feature in your build, but that's pushing a little past your budget unless you find a good sale or used one.
Even at the top end (RTX 4090) performance difference between PCI 3.0 and 4.0 is pretty negligible.
Yeah, it's at most a 10% decrease, but didn't want to mislead on it.
I would look for an RTX 20xx series GPU and then upgrade the RAM if you can.
I’d be looking at a used 2070/Super or 3060/Ti in your situation.
If you’re on Windows, consider that you have a year of support left with Windows 10. You can buy further security updates from Microsoft, from 0patch, or with an “unsupported” Windows 11 upgrade. There’s also Linux if the games you want to play are supported :)
With that in mind, you can consider the alternative of limping along with what you’ve got and saving for the next year to do a more substantial upgrade / overhaul.
Whatever you do, make sure it has more than 8Gb VRAM.
People said the 3060 would outperform the 3070 in real life usage because it has more VRAM and I’m still kicking myself because they were right.