Try unplugging all add-ons and boot with the PSU you normally use?
I don't think there's a great deal of risk of toasting your A1200 because of the PSU, unless the PSU is smelling odd, making unsavoury noises or getting especially hot (if you could burn your hand on it by holding it there, that's bad).
I take it you're not in the habit of unplugging the PSU from the A1200 that often? It might be worth checking connections inside the A1200 where the PSU plugs into the motherboard, IIRC that wasn't the sturdiest of interfaces.
As a precaution anyway you might want to unplug your hard disk until the problems are sorted, because if the Amiga is power cycling quickly, the hard disk heads will be going ape.
