If you are moving from an 8700K (with a 1080ti) like I did (with a 1080) and you're using a 7800XT now (I went for 7900XT)... I would simply just get a much faster CPU. Save a bit more and get an X3D; even a 5800X3D will be a great move - and you can run that on DDR4.
And if budget doesn't allow for a solid move forward, you know what? Just sit on your current hardware a bit longer. Stuff gets discounted. Keep your eyes open and wait for a killer deal on stuff you actually get a real advantage out of. Cheapskate upgrades are no upgrades; you're already in high-ish end GPU territory here and the CPU is no slouch to begin with, its just too slow for your GPU. Again, I did the same thing, I waited until I felt really comfy burning a bunch of cash on a new platform. You can make do with less, but do it smart; reuse the RAM, make it a best-in-class upgrade on DDR4 instead. 5800X3D is literally the fastest gaming DDR4 CPU you can get.
ALSO, consider finding a new home for your 8700K based stuff minus the RAM, or even with the RAM if that makes the deal work for someone else, that's your missing budget right there. Its still a valuable CPU, really.
IMO I think that money would be better spent on a 1440P+ monitor. The 8700K is old but it's not cripplingly slow, it's about the same as a 10600K at the same freq for games which is totally fine for 1440P/4K.
Naaah its definitely long in the tooth, I've been using it with a 7900XT for a few months and it wasn't as pleasant as you might think. If you get a current gen 7800XT or better, you need more CPU for it to be a balanced system again. Its a fine CPU for something maximum like 6700XT and that's about where it stops. I had a 40% bottleneck on the 7900XT in some benches, that's pretty heavy