The fuel pump should not be running with the key off,it sounds like it is wired to the wrong side of the ignition switch. If there is power to the pump with the switch off then there is a problem so you will need to trace the wire from the pump and see where it is hooked up to and if its hooked to the battery side of the ignition switch you will need to hook it to the ignition or acsessory side or you might have a defective ignition switch also or someone hooked it to a constant hot wire. I ran a electric fuel pump on a 53 olds for many years and I had it hooked to the ignition post of the ignition switch and had no problems. Jeff