I don't think there are any FM2 chips that will help.
If you need to change your motherboard, then there are intel options like the i5-4460 in that price range.
http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106-3.html
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 could also experiment with removing one core. You can do this in the windows msconfig boot advanced options option. 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.
Many games would perform better with a overclocked $65 G3258 or a $110 i3.