Wowa. The fuel pump is only supposed to run for a brief moment when you turn the key to on. And then of course while the engine is running. In no way should that fuel pump be running with the key off. I hate to throw parts at the thing, but the fuel pump is sure suspect. There is a hatch in the floor to change it. My thought is vapor lock, but for a fuel injected vehicle there has to be a cause, ie: a fuel delivery issue.

1996 SR From the Area 51 Skunk Works. 37"BFG Tires, 5" lift, Rock Track 4:1 Case, Tom Woods Drive Shafts, Oasis Air, Front ARB, lifted gas tank, T-max winch and lots of stuff that cost too much.