From what I am seeing online, my proc uses an FM2+ socket, and the one I would like to replace it with uses an AM3+ socket. Would that work, or not?
Also, would my computer be able to handle that without overheating?
No, your computer has a motherboard that has socket FM2+ and requires a CPU (or APU) that is socket FM2+. It cannot use a socket AM3+ CPU.
So you can:
A. upgrade to something like the A10-8850K socket FM2+ APU which will have similar performance plus another 200 MHz (maybe not worth it, depending on price/availability of a faster APU)
B. If you already have the FX-8320, replace your motherboard with a socket AM3+ board (might be economical if you can find a used motherboard at low cost, but it is a system rebuild.)
C. Rebuild with a Ryzen based system (costlier, but better performance now and into the future).
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