If you can test the fuel pressure when you get it running that would be best, just attach a guage to the t-valve output with the t-valve open and turn on the truck to see lift pump output. The very least you should be able to open the filter bleeder when its running and have fuel shoot up an inch or two.
When the truck doesnt have oil pressure but you are cranking the relay powers the lift pump, but as soon as it fires up and oil pressure comes up the OPS take over powering it. So that might be why it would die after 5 seconds.
Its also a quick way to kill the IP, so you want to verify the LP is working.
I dont know if it not staying on after you shut down means its bad, but ever since I changed my OPS and LP I notice it a lot more. You'll need a certain socket to replace the OPS if you do that
KD tools makes the socket… socket # K-D 3458. it is 3 3/16 inches overall length, 7/8 hex head on top with 3/8 square drive hole for rachet, extensions and so on.
OPS # OP6677 Napa / Echlin brand
Not sure if there is any difference in the OPS and lift pumps but the ones on SSDiesel say they are Heavy Duty, but I imagine any you buy anymore would be the same.