There is definitely something wrong with the one piston. Exactly what, is difficult to say, but I would go with- 1) Stuck rings (either carbon fouled or overheated-stuck) 2) Broken ring(s)
I have fumbled with trying to use ring-free type products to loosen some seriously stuck rings on both an 85 and a 90 hp OMC, but neither case proved successful. IMO, a tear down is the best solution.
In your outboard's case, remember that the powerhead is almost 20 yrs old. That's potentially a lot of time on the old girl, so you will not be throwing away just like brand new pistons.
As to the cutting out... That could be related to the one bad piston. At particular rpms, the one bad cylinder could be causing some exhaust resonance problems.
If your oil pump was cutting out, you would have far worse problems than overheating. Overheating- check water pump and cooling systems.
If you do doubt your oil pump, be sure to run premix until you either become confident or find the problem.
Good luck with it.