Good (simple) check / suggestion and nothing to lose. ;-)
I believe that pot is just a trimmer allowing you to tweak the voltage around the nominal 12V so I don't even have to mark it and would re-set it under load.
I'm not particularly hopeful it will do anything as even if it was 'noisy' and was allowing the voltage to fluctuate around it's entire range, I would have thought it should still have been able to drive the 5A load, even if also whilst fluctuating? I wonder though if that only depends on if the PSU still works at all with (worst case), no voltage on the wiper of the trimmer?
I'll let you know how I get on. ;-)
Cheers, T i m