To answer your question:
You need a better than average motherboard to overclock a FX-8xxx as well as a better than stock cooler.
What is your motherboard.
Other cpu options will depend on how well threaded your games are.
Typically, sims as well as strategy and mmo games depend on the performance of a single master thread. Unfortunately, FX cores are weak.
Here are some tests to give you some insight on this:
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 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.
Intel cores are some 35% faster per clock than FX.
My guess is that a I3-6100@3.7 with a better single thread rating would do you some good.
The I5-6400@2.7 not so much; likely a sideways move.
If you will change out your cpu and the requisite motherboard, go with latest 7th gen.
I3-7100@3.9 would cost no more.
Your absolute best would be a I5-7600K with an overclock.
As of 1/13/17
What percent of samples can get an overclock
at a vcore around 1.4v.
I5-7600K
5.1 28%
5.0 52%
4.9 72%
Ryzen official benchmarks are not out yet.
The suggested 52% IPC improvements might make such a chip worthwhile if true, but it will come and a higher budget than you are suggesting.