It sounds like signal reflections, probably caused by incorrect line termination. A composite video signal should be terminated at both ends by a 75ohm impedance. Some cheap TVs seem to have issues with correct termination somewhere in the signal path, in this case most likely in the signal multiplexer which is used to switch between the video in and the internally generated video.

If the signal isn't correctly terminated, sharp changes in level (black to white transitions, sync pulses, etc) which have a very high frequency component and lots of harmonics will reflect from the driven end of the input and bounce up and down the cabling until the energy is dissipated. It's also called 'Ringing'. I would guess that in this case the guts of the thing are pretty cheaply made (it IS a Zenith!), and the input amplifier simply isn't very high quality. Possibly the termination resistance, whatever it is, is changing value as it ages, or it could be reflections caused by bad PCB layout in the imput stage. In either case, there probably isn't a lot you can do about it.

That said, have you tried a new video cable? It's barely possible that there is a problem with the impedance of the connectors on your current one, or something of that nature.

pca