I hate to be the guy who says it, but: just replace all the capacitors!
Really, the buzzing sound is likely a bad filter capacitor. You could probably start by replacing only those. Do not keep it plugged in for more than a few seconds to verify the concern!
I have a GE console stereo as well, it's one of the few large item trash picks I got when I was young. The cabinet of mine is a little different - more modern styling, though probably of roughly the same vintage. Mine worked fine straight out of the garbage, but that was over 10 years ago, when even older radios from garage sales and such seemed to work right off the bat. Nowadays, they're so old that there's always something wrong with them, just due to age. And even though the (more professional) vintage repair guys will try to tell you to pinpoint the problem rather than using the shotgun method (replacing ALL the capacitors), the caps are all old. We're talking 70s, right? That's 50 years ago. 50 years is a long time for those little wax paper pieces of crap.
Tubes are actually pretty robust technology, and very rarely go bad... other than being worn out with use. However, running a set that has bad caps, or any other components really, for more than a minute or two could easily damage the tubes.
Not sure if that's a solid state set or not, anyway.
I use digikey for caps and any other electronic parts.