For most games, the FPS difference between the FX4300 and the 8320 is next to nothing. You'll see some boost, but nothing much. Meaning if you are getting ~30FPS now, moving to the 8320 won't get you 60FPS.
http://www.techspot.com/review/586-amd-fx-8350-fx-6300/page6.html
While the 4300 isn't on there, the older (and slower.) 4170 is. In Crysis 2 the 8350 gets 35.4FPS, while the 4170 got 33.1. So like I said, you'll get a boost but nothing worth spending ~$100 on. I agree with SR-71. It's time to upgrade the machine. New board, CPU, RAM, etc.
Edit: Here is a better, newer review.
http://www.hardwaresecrets.com/fx-4300-cpu-review/8/
Notice in some of the games they are all basically the same. (BF4, Dying light, etc.)...