I think the assertion that AMD 8 core chips will be needed for next gen games is AMD FUD.
No game developer will have an 8 core requirement, or they won't sell many games.
The link shows that a dual core i3-3245 is ok which tells me that a quad will be plenty.
For recommendation, I see a 3770 which makes no sense to me when there are better chips available at a lesser price.
High/ultra settings are the province of your graphics card, not the cpu.
Buy a strong graphics card first.
If you have concluded that your current cpu is not up to the task, then ask why not?
To help clarify your CPU/GPU options, run these two tests:
a) 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 could also experiment with removing one core in the bios. 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.
It is possible that both tests are positive, indicating that you have a well balanced system, and both cpu and gpu need to be upgraded to get better gaming FPS.
For the cpu, look at the latest tom's best gaming cpu's report.
http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106.html
Only the FX-6300 finds a place at $110, all the others above and below are intel.
If your motherboard supports a FX-8350, then you could upgrade.
But, all you get are more cores which do not help gaming.
Actually, your X4 965 is very decent for gaming. I am told that less expensive am3+ motherboards do not do well with FX-8350.
If you need to replace it, then intel will be much better way to go.
Here is an older report on <$200 gaming cpu's.
http://www.tomshardware.com/reviews/gaming-processor-frame-rate-performance,3427-9.html