We just have to figure out which signal is staying on. Here is some info about the fuel pump relay.
When the ignition switch is turned ON, the Electronic Engine Control (EEC) power relay is energized, providing power to the fuel pump relay for 1 to 2 seconds through the Powertrain Control Module (PCM) and an Inertia Fuel Shutoff (IFS) switch.
Fuel pump power is provided through the fuel pump relay contacts, if the PCM does not receive an ignition signal within approximately one second after the ignition switch is turned to the "ON" position. A timer circuit in the PCM will open the ground circuit to the fuel pump relay, causing the fuel pump relay contacts to OPEN. This action stops the power supply to the fuel pump.
When the ignition switch is turned to the "START" position, the PCM closes the ground circuit to the fuel pump relay, resuming operation of the fuel pump.
After the ignition switch is returned to the "ON" position, power to the fuel pump is supplied through the fuel pump relay.
The PCM monitors engine speed and opens the fuel pump relay ground circuit if the engine speed drops below 120 rpm.