Sounds like the fuel pump relay in under hood fuse box try finding another next to it thats the same from the headlamp relay and swap them and see if it helps. I suggest you check the net for info on how a fuel pump relay works and conduct the electrical test.
6 Signs Of A Bad Fuel Pump Relay And How To Test For A Failed One
Fuel pump keeps running with key on. So i ran into a little problem with my return style fuel system. That makes sense to me im going to try that tomorrow. If the pump runs with the relay removed its not the relay. Replacing the fuel pump relay stops this from happening for a short time but it soon reoccurs. If it is stuck the pump could continue to run. Once you located open the fuel door and take off the cap and leave it open.
It sounds like you have a short to voltage somewhere. Turn the key to the on position without starting the engine. You should hear the fuel pump running for 5 seconds if working. Start the engine take off the fuel pump relay and leave the engine stop itself when this occurs turn off the key the you can take off the filter and changed it. If not heard running check the oil pressure switch for the 12 volts needed to run the fuel pump starts there on that harness. A full explanation of how the relay works and how to check it is quite long.
If everything works correctly key shuts motor off kill switch shuts motor off and fuel pump runs when its supposed to then its the alt. To do that first find the fuel pump relay in the main fuse box located in the engine. It wont turn off and it keeps running even with the key is off. This primes the gas to the injectors when doing this. The fuel pump on my 1989 mercury grand marquis is still running after turning the ignition off causing the battery to drain. Is it possible there is a short in the ignition system or is this a sign that the fuel pump is going out.
Normally the fuel pump relay operates during starting and when turning off the engine.