[Eliot Ness]

We're not out of the woods quite yet, though. I let the radio play for about 30 minutes, and when I went back into the room after breakfast, I was met with a very powerful smell. Burning plastic or something. It was still playing fine, but the whole room reeked of that burning smell

Like the others have said about the caps and resistors.  Also look for signs of overheating like discolored components and tar oozing out of power transformer (under chassis), etc.  Sometimes your nose is pretty good about localizing the smell (power off please!), other times it will be hard to narrow down after the odor dissipates.  Good luck!