It's cheap and easy to test a battery, it may be worth while before taking the truck to a mechanic. Like I mention, a new battery did fix the problem at least for a couple weeks. I'd had so many problems with that vehicle I wasn't convinced that just replacing the battery was going to fix the problem so I went ahead and sold it while it was running. I couldn't understand how a battery would crank the engine but not supply enough juice to keep the engine running, the shop insisted the alternator was working fine and the battery just wasn't holding enough charge. Replacing the battery solved the problem, at least for the two weeks I drove it before selling it. I'd already replaced the fuel pump and relay previously as they'd gone bad 6 months earlier. The engine would crank and start, it would even run but the gages would bounce around and acceleration was sluggish, like the fuel filter was clogged even though it was only 6 months old. My Chevy Blazer with a 4.3L (same as a 350 minus 2 cylinders right?) showed similar symptoms before I got rid of it, I took it to a shop and all they did was replace the battery. There are a few others here whose opinions I would trust. I have not been actively working on cars for over 10 years. Look for cracks, signs of arcing and moisture. Does the car actually have a distributor? If so, you could take the cap loose and check the underside of the cap and the rotor. If you think you could have bad plug wires, raising the hood in the dark with the engine running will allow you to see any arcing. Obviously you want to be dealing with the correct one. Even if there is just an idiot light, there may be two switches. If the truck has an actual oil pressure guage, the sender for that may not also function to cut the pump off. Worth investigating, but without a guage to check pressure, replacement will be your only option assuming the wiring is not suspect. You most likely are not loosing oil pressure per se, but instead the oil pressure sender could be going bad. If the engine loses oil pressure the fuel pump will be shut off. Swap the fuel pump relay for another one to see if the problem is corrected.

Find where the fuel pump relay is located and if it's in an underhood relay center, there will be other relays there. One thing you could do is try a different relay. It's also important to know what the correct pressure should be. There is a procedure to follow that can tell you whether it's the pump, the regulator or even a blockage in the lines.

There is more to it than just checking the fuel pressure. I own one, so as far as renting I don't know.

Pulling cam 350 chevy