1998 Chevy Silverado 5.7 Vortec Fuel pump question

Hello all, 
When cycling the key on my truck, the fuel pump primed for 2 seconds.....then when I turn key off it will prime again for 2 seconds. Is that normal? The fuel pump will also run for a few seconds after shutting the engine off. 

To my knowledge, a second priming when cycling the key off is not normal behavior.  Does this happen even if you do not start the engine? ...just turning the key on and after the first 2 second priming event, the fuel pump shuts off (as it should), leaving the key on for a couple of extra seconds without starting the engine and no fuel pump running? Then, turning the key off initiates a second priming? I do not believe that is normal. Does it run for the exact same amount of time, every time. Or, does the run time vary, slightly? Maybe, 2 seconds this time, but only 1 second, next time?

If it only does this after the engine has been running, I would suspect the oil pressure switch. 

As a backup strategy for a failed fuel pump relay, the oil pressure switch will, also, supply power to the fuel pump as long as there is oil pressure. In the event that the fuel pump relay fails, cranking the engine long enough to build oil pressure will close the oil pressure switch and energize the fuel pump. To test for a slow-to-open oil pressure switch, disconnect the oil pressure switch and see if the fuel pump, still, continues to run after turning off the engine. 
 

That's a really good point about the oil pressure switch.
I can remember some old Chevy vans, while performing injector drop testing, that the fuel pump would every once in a while prime key on, and then prime again key off. Not consistently, but it was something I noticed (15 years ago) and forgot about until reading this post. I don't think I have ever encountered one continue to run after shutting down the engine.
I like the oil pressure switch circuit for this one, I think Chad nailed it.

Hello Guys, the fuel pump primes exactly the same time on and off with key. First thing in the morning, or after driving it. I've been chasing a stumble when starting it as well. Fuel pressure on new MPFI is 55 key on, and doesnt lose pressure. And running it's 47-48....new pump and filter, new relay, new ignition module and coil, went back to OEM after trying it, and new Delphi MAF sensor. 02 sensors (NTK) are new. No cats so only replaced the front two. No codes as well. Has a new distributor, cam, crank, sensor, and All AC Delco, or Delphi, or GM Genuine. And AC Delco distributor cap and rotor.

But one thing I did while I was back there changing the distributor I put a new Standard Motor oil pressure switch. And I don't know if it's went bad causing this.

And yes I replaced all ground straps and checked ground at pump. I checked current flow on relay jumping it and had 4.5 amps of current. Which seems low to me....

New harness connectors as well. Including a new GM genuine Coolant sensor near the thermostat housing

Fuel pressure on new MPFI is 55 key on, and doesnt lose pressure. And running it's 47-48....

Fuel pressure spec for that truck is 60psi-66psi

I own a 98 silverado with the 5.7L and have owned it since new and mine does the same thing as yours does. Primes for 2 seconds when turning the key and then does it again when shutting the key off.

Mine has a replacement AC-Delco fuel pump that was installed probably 20 years ago or more.