Not staying running
From what you have posted, it sounds like you have already verified you have power in the yellow wire at the fuel pump relay all the time.
Have you checked for power coming through the oil pressure circuit to the fuel pump relay?
Again looking at the wiring diagram, with the key in run you should have power through the 15 amp fuse to one of the wires (R/Y according to the diagram) at the oil pressure switch. With the key in run verify you have power at the oil pressure switch.
Now, the purpose of jumping the terminals of the oil pressure switch is to deliver power to the fuel pump relay. So once you have verified you have power at the oil pressure switch (with the key in run) you should have power in the R/Y wire at the fuel pump relay if the oil pressure switch is jumped correctly.
If you have power in the Yellow and R/Y wires at the fuel pump relay, and you know the relay is good, then the only thing left is the fuel pump relay ground wire (black according to the diagram) which you can check with a test light.
The only other thing I can think of is the possibility you didn't actually change out the fuel pump relay but changed the tank selector relay instead? I don't have a tank selector relay in my vehicle but I have seen it said they look very much alike.








