I had an old Bronco II that did that when I was a kid. I didnt know how to fix it myself. My Grandpa was able to fix it for me, but I wasnt there when he did it. He told me the problem was a bad fusible link.
Im not sure where the fusible link was located, but I'm guessing it was in line to the pump somewhere and easy to overlook.
Not sure if this info will help you.
2012 Mustang Cyclone, 2002 Mustang GT, 2004 F150 FX4, 1999 Ranger O/R 4x4, 2007 Harley FXDB