You will get little benefit from a FX8350 or such.
Your problem is that FX cores are slow, and adding more of them will not usually help.
Most games can use only 2-3 threads effectively.
Test this with YOUR games.
Experiment with removing one or more cores. You can do this in the windows msconfig boot advanced options option.
You will need to reboot for the change to take effect. Set the number of processors to less than you have.
This will tell you how sensitive your games are to the benefits of many threads.
If you see little difference, your game does not need all the threads you have.
Time to save up for a modern kaby lake or ryzen processor.
Even , a $65 G4560 will usually do better than your FX-6300