The point is to not discount the PSU as a problem, it is very important. Most prebuild systems have very poor PSU, and google a little and you'll see your's too. Beside that, 350 is rather low, even for a quality PSU, so if you're upgrading, that should be on your list. Replacing a PSU is not expensive not difficult (usually)http://www.newegg.com/Product/Product.aspx?Item=N82E16817139026
$25 after $20 rebate
Then I'd say you can go 7770 maybe more depending your CPU (which is rather old), but you're safe knowing worst case is a bottleneck not a neckbrace (being facecious). And you can reuse a good PSU for your next build.