It could be the fuel pump relay is bad. It is the center one of the 3 mounted on the passenger side fender. Here is a link showing it's location.
http://www.gnttype.org/maint/sensors.html
Here is some good info from vortexbuicks-etc.com
Factory Fuel Pump Relay
Contrary to popular belief, the factory relay does not shut down the engine if the oil pressure drops, or is lost. The factory relay is backed up by the factory oil pressure switch which has two functions. First, the switch turns on the oil pressure idiot light if the oil pressure falls below 4 psi.
Secondly, the factory oil pressure switch completes the circuit to the fuel pump when the oil pressure rises above 4# as a back up in case the factory fuel pressure relay fails. Normally, the ecm signals the relay to start the pump and run it for about two seconds when the key is switched to "Run", then it cuts off if the engine is not started. If the relay fails, there would be no fuel delivery from the pump unless there was a back up circuit. This is the second function of the oil pressure switch. Even if the relay fails and one cranks the engine, the fuel pump will be turned on when the oil pressure reaches 4 psi during cranking. One of the signs of a factory relay failure is delayed engine start when the engine is cranked as it gets no fuel until oil pressure is above 4 psi.
Now, if the factory relay has failed and the engine suddenly loses oil pressure to below 4 psi, then the voltage to the pump would be cut at this point and the engine would stop...that is the only time.
