Powers off after 30 secs, Permanent damage?

Reseat ram, all plugs, video, anything that could come loose.

If you have it all setup in a new motherboard with a 20 pin connection and everything is working, then I would say that it is your motherboard that is the problem.

If your PSU was powering it previously, and is still working on the new board, then something has happened to your old motherboard - either you damaged it or something wasn't seated right (came loose during the trip).

Did the manual say it was ok to run the board on a 20 pin power connection?

I'm more inclined to suspect the PSU. A power-off after 30 seconds sounds like an overheating. You say you moved; very likely there is a difference in the locally supplied voltage at the AC outlet, e.g. 114, whereas the old house had 125. To have a 20-pin connector suggests it is also an old PSU, even if it was once good. What brand and model is it?

Much of the time, a PSU that comes with a case will be crap. Do you know the brand and model? The fact that it is powering off rather than blowing up suggests it at least possesses some protection circuitry, but I would definitely not keep tempting fate by using it.
Yes, you can plug a 24-pin PSU into a 20-pin mobo. The extra pins are some extra +12V lines. Going the other way (20 pin PSU into 24 pin mobo) can be dicey because those additional lines are to support a higher current draw that may exceed the rating of the wires themselves. You could in fact damage your 24-pin mobo if a wire overheats at the connector and something melts.