12600K is more than enough for gaming, you won't gain much going up.
If money is an issue you can get the non-k 12700 or 12700F. Few games will take advantage of more cores, with fewer faster being better. The extra cache on the 12700/12900 doesn't help much either.
Getting much above 5ghz is harder anyway, and all turbo to around 4.9 give or take.
I've used a 12600k and a 12900k. In normal desktop usage and games, I wouldn't say there was a noticeable difference. There are plenty of reviews out there to show you exactly how much difference you'd expect to see.
As an owner of a 12700K who mainly plays games I would go with that. I've read good things about the 12600K as well as a gaming cpu, so in reality you can't go wrong with either. I will say you'll see quite a performance boost from your 9600K.