"My guess is that the new PSU is giving the processor too much power and it's overloading."
That's not likely, your BIOS settings will determine the processors voltage.
"I get a beep code that seems to mean processor overheating."
Assuming that you've got that correct, try a new CPU fan, your's might be failing, also make sure all vents, fans and case internals are clear of accumulated dust and fluff. If there are any case fans fitted, ensure they are running properly.