The answer will depend on the types of games YOU play.
Most games are dependent on the performance of a single master core.
I might suggest a $126 i3-6100 and any lga1151 motherboard. The single core passmark rating is 2102.
It is dual core with two added hyperthreads.
Your FX-6300 is actually triple core with some very good hyperthreads.
It has a single thread rating of 1408.
ASROCK has announced non "K" overclocking on some new motherboard.
You might look into that.
Here are some tests you can make.
) Run YOUR games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
b) Limit your cpu, either by reducing the OC, or, in windows power management, limit the maximum cpu% to something like 70%.
Go to control panel/power options/change plan settings/change advanced power settings/processor power management/maximum processor state/
This will simulate what a lack of cpu power will do.
Conversely what a 30% improvement in core speed might do.
You should also experiment with removing one core. You can do this in the windows msconfig boot advanced options option. You will need to reboot for the change to take effect. Set the number of processors to less than you have.
This will tell you how sensitive your games are to the benefits of many cores.
If your FPS drops significantly, it is an indicator that your cpu is the limiting factor, and a cpu upgrade is in order.
The best <$ 200 quad might be the i5-6400@2.7.
Note that the i3-6100 clock is 3.7 so it pays to know what you actually need.