Oil switch is pretty cheap here, put it in.

As far as the pump relay being de-energized after oil pressure builds, I've
never seen any published information, stating that. I've worked on alot of
GM vehicles and I have never seen that.

There should be a fuel pump prime connector. You can run B+ to that
connector and the pump should run. You don't have to use the key, you
don't have to worry about the oil pressure switch in the fuel pump
circuit, you can try this after the vehicle warms-up and dies, just turn
the key off.. This is just for a test, you can see if the pump runs and
check fuel pressure. I didn't notice a fuel pressure spec listed in your
post, maybe I missed it.

If the system was holding oil pressure for a long time, that o/p switch in
the fuel pump circuit, would remain closed, meaning the pump would run,
until either the oil pressure drained off or the key was turned off.

If there is a red wire coming off the relay, that should lead to the fuel
pump prime connector. That connector is usually on the same side of the
engine compartment as the battery, that doesn't mean it is close to the
battery. The connector can be in different locations on different models.
Sometimes their close to the relay. You could take the pump connector
loose, then run B+ to the prime connector and check for B+ and ground at
the pump connector. Usually you can get to the pump connector without
dropping the tank, you'll have to look.

Like TED suggested, your pump may be having a problem.

Are you saying, according to the manual,once the oil pressure rises, the
fuel pump relay is de-energized?

I just looked at a wiring diagram on your vehicle, the computer has no way
of knowing what your oil pressure is doing and no connection to that o/p
switch in the fuel pump circuit.

I have an old 88 GM vehicle. I have two oil pressure switches, one is
located by the oil filter, that is for the lite or gage on the dash. The
other is located on top of the engine, that o/p switch is in the fuel pump
circuit. In the past I have unplugged that switch, the vehicle started and
ran just fine, the pump was getting voltage through the relay. I plugged
that switch back in and unplugged the fuel pump relay, just let the
connector dangle. The vehicle started and ran just fine. The pump was
getting voltage through the oil pressure switch.

How old is the fuel filter and what is the fuel pressure spec. Maybe you
listed fuel pressure and I missed it.
My generic info for a GM TBI system, says the pump should draw 2--4 amps.
If you check amp draw using the prime connector, I'd use a fuse on the
circuit.