I posted a build above with a 7770 and a Phenom in your budget. Unless that $12 is a big deal, in which case I can see what can be done.
You can always improve on stock fans, of course (mediocre stock fans is a universal phenomenon). We're not talking about chainsaws in your computer room - on a tight budget, you have to make significant compromises. Me, I'll prioritize significant performance gains over a few decibels of noise, but you're free to disagree, of course.
The Phenom IIx4 965 being a staple of entry-level build proposals here and the Athlon IIx4 rarely if ever featuring isn't done for insignificant reasons or caprice. It's a better CPU.