With that in mind, change the fuel pump relay under the hood with another of the relays, which are identical; I suggest A/C but be sure they are the same number - they are little black plastic cubes. Just swap, and see if it helps, at no cost for this diagnosis.
Our '99 was doing this, and it was the ignition key lock cylinder, the tumblers were sticking, or springs were weak in there, but that's $400.
