Have you looked at your gpu/cpu usage to see whats really bottlenecking? Monitored all your temps to see if your thermal paste has given up and you're thermal throttling? The 3600 is older but not a slouch, it should be fine, and I'd also expect a 5700xt to be doing much better than you describe unless you're running at 1440p/4k native. It's not that old.
Use software like HWmon, Afterburner, or HWinfo64 (sensors only) to check your max temps on CPU and GPU cores while gaming.
Also consider doing a complete wipe and fresh reinstall of Windows 10. I always have to do that every 2 to 3 years to keep things running fresh because it somehow always turns into a mess.
Upgrade paths:
You can easily drop a zen 3 5000-series cpu into your current motherboard as long as your BIOS is updated. The Ryzen 7 5800x3d is considered the second best gaming chip of all time currently, and can be had for around $300 right now. That alone will easily, easily handle pretty much all games for at least the next 5-8 years. It's seriously powerful.
On the gpu side.... life is weird, and will continue to be weird. I cannot recommend any of Nvidia's offerings in good conscience just because of their monopolistic tendencies. If you're trying to stick around the 5700xt's $400 msrp, your options are going to be either the RTX4060Ti from Nvidia, or the 6700xt/6750xt from AMD. Both of which trade blows in most 1080p games, but the AMD card comes in a little under $400 since it's older and the 4060ti will run over $400 because actual msrp models are often scarce and Nvidia is a monopolistic greedy fucking corporation. However with the Nvidia card you get shit like DLSS and a little bit of raytracing (although acceptable RT performance in a 60ti card is still very much NOT there).
AMD has yet to launch their ~$400 offerings in the RDNA3 7000 series lineup, although they're supposed to be coming "soon™️". So if you're OK with waiting a few more months, the 7700xt could be quite attractive.