I have a TDS350 which is similar and the problems I had were similar to yours: 1) the baselines would shift around, 2) the traces would vanish off the screen at 50 mV/div and below, and 3) there was approximately 7 volts on the input BNC connector. Incidentally, as soon as I put a load on the BNC, the 7 volts dropped way down. All this happened only on channel #2 in my scope.
My conclusion - and I know this isn't good news - was that the input preamp hybrid was bad. I managed to find one fairly cheap on eBay and that fixed the problem. The hybrid doesn't look like a circuit that is easy to repair - there is all sorts of buried stuff if I recall. Perhaps someone else on the forum that has actually fixed one successfully can chime in.