I did some more testing. I limited the testing to just one app, the stock camera app.
Doing a dummy video did nothing. The same sync problem occurred at the pixelation error at about the ten second point each time. (I earlier said that the issue occurred at the one minute point. I don't know what I was thinking. It appears always to occur at ten seconds.) I think that maybe the phone is having trouble moving video and/or audio data to memory fast enough and whatever it can't move it drops. And this dropping of data is causing the sync problem.
I killed all unnecessary applications using Advanced Task Killer and the sync problem disappeared. And when I say all unnecessary apps, that's what I mean: nothing except camera and Advanced Task Killer. Maybe I can narrow it down to one app that's causing a problem, one app that would be common to all stock Droids, like Voice Recorder or whatever.
I suspect that if I overclock the phone to maybe 750MHz the problem will completely disappear. So I'm going to try another ROM and overclock it and see what happens.
But anyway: success with killing all unnecessary applications.