Seems that the analog board in the Classic in particular is vulnerable to capacitor issues... This thread documents it somewhat: http://68kmla.org/forums/viewtopic.php?f=7&t=21471

I haven't seen the exact issue you're describing on a Classic though, myself - I do have an SE I'm trying to figure out that has a fuzzy CRT, and when I changed the analog board it started wobbling. From what I've read so far, it could be a power supply issue (caps in there of course), flyback transformer, or analog board caps.

Correct me if I'm wrong, the Classic has the power supply built onto the analog board, right? So the caps there are for the video circuitry and power for the whole machine?