The FX-8350 may cause bluescreens and instability. An FX-83x0e would be a safer bet, they have lower TDPs and should not cause issues.
Be aware however that even an FX 8-core is generally no better than a Core i3 in games, and is both more expensive and more power hungry. I'm aware that you'd need to change boards to move to an Intel chip and I'll not try to convince you of that, but a better solution to your problem might be to offload some of the streaming and encoding work to the video card using Shadowplay, rather than investing in a very old, slow, and power hungry CPU.