I just got an LCIII from ebay and it has that whine/hiss/am radio noise too.

From my hardware background, I would think it was a failed capacitor too. I've noticed that while it oscillates audibly, the actual sounds going over the speaker (Apple Menu -> Control Panel -> Sounds, then clicking an alert sound like Quack) will be garbled. I would suspect the problem is isolated to the audio driver. Maybe something like a 555 Op Amp or a supporting capacitor.

Anyone isolate this further? If so, please post! I will post what I find.