I don't have Flash benchmarks, in fact I don't have 6320 benchmarks
but when the 6310 is capable of playing Modern Warfare 2 I would expect it to be capable of Flash as well.
http://www.anandtech.com/show/4023/the-brazos-performance-preview-amd-e350-benchmarked/5
On the other hand, Firefox has been hardware accelerated since 4.0 http://www.mozilla.org/en-US/firefox/4.0/releasenotes/ (and yes, your GPU is good enough for hardware acceleration) and has improved every version. Not only that, but using an old browser is asking for viruses. From what I can tell 3.6 will EOL on April 24th, meaning no more security updates (though if you think it's secure as is you're kidding yourself.)
Upgrade already. Free instant joy, assuming your comp is secure otherwise.