A RAM upgrade to 16GB might help, I would try that first. Worst case scenario you can use that RAM for a new build of you're still not be completely happy with how it performs.
Streaming definitely affects gaming performance but your specs seem to be okay for 1080p, depending on which games you play.
Do some monitoring using Windows Task Manager to see if buying more RAM will help.
A 650w or 750w would be ideal. If you can't afford a PSU upgrade right now, run your CPU at stock speeds and don't run any GPU overclocks and you'll be fine, but I would be looking to upgrade soonish. 80+ Gold PSUs seem to be much more affordable these days so I would consider one at least 650w to deliver plenty of clean power to your new 1080!