It's possible that the ASD relay is at fault. Here is some information from ALLDATA about it that may aid you in checking it out.
The PCM energizes Auto Shutdown (ASD) relay by supplying relay ground, when ignition switch is in START or RUN position and PCM is receiving a reference signal from distributor.
When the ground is applied, the relay is energized, and voltage is allowed to flow to fuel pump, fuel injectors, ignition coil, generator field winding, and O2 sensor heating element.
When PCM does not receive reference signal from ignition system, (indicating engine is not running), then the PCM interrupts relay ground circuit and no voltage is supplied to fuel pump, fuel injectors, ignition coil, generator field winding, and O2 sensor heating element.
The PCM controls fuel pump relay and ASD relay simultaneously, through same ground circuit.
The Automatic Shut Down (ASD) relay is located in he Power Distribution Center (PDC). Power for the coil side of the relay is provided by circuit F12. This circuit is HOT in the START and RUN position, and is protected by a 10 Amp fuse in cavity 18 of the junction block.
Power for the fuse is supplied on circuit A21 from the ignition switch. The A21 circuit receives power from the A1 circuit. The A1 circuit is protected by a 20 Amp fuse located in cavity A of the PDC.
The Powertrain Control Module (PCM) controls the ground path for the coil side of the relay on circuit K51. This circuit connects to cavity 51 of the Powertrain Control Module (PCM) connector. It is also spliced and connects with the coil side of the fuel pump relay
When the ASD relay is energized, the contacts internal to the relay close connecting circuits A14 and A142. The A142 circuit is spliced and provides power to the fuel injectors, generator, and the PCM.
Circuit A14 is protected by a 20 Amp fuse located in cavity B of the PDC.