Stuck in Cullman, Al 2001 Saturn SL1
 

My SL1 runs for awhile then hesitates and then runs okay for a very short period. I can hear the fuel pump surging. I just replaced a couple day ago the fuel filter. Could I have a bad filter or is the fuel pump going out?

Normal fuel pressure is about 55psi with the engine running

Checked the fuel pressure. It runs about 50 psi, then drops to less than 20 psi and comes back up. Doesn't go to zero. Oh, that's with the engine running.

You're in the ballpark. Fuel filter (replace it again) ... fuel pressure regulator ... or even the fuel pump. Start cheap, and work your way up........

The basic back yard rule of thumb is if you can blow through a fuel filter, it ain't blocked. If it is blocked it generally does not change the amount of blockage.
While most of my cars that have quite because of fuel pump quit because the pump quit.
It usually did not surge like that, but if you can hear it doing it, it is. I have had a pick up problem in the tank where when the tank was low it was an intermittent problem but that car was converted to fuel injection and the pump was external and the tank really was not designed to be used that way.
It sounds to me from your description the pump is on it's way out.