There is a "relay" for the fuel pump in the power distribution center, the black box under the hood on the passenger side.
Before you install that new fuel pump, you might try swapping the fuel pump relay with a non-critical relay and see if that resolves it. These relays are all the same part number so no issue with swapping.
If a relay swap takes care of your problem, then replace the suspect relay.
99 Cherokee, 4.0 AW4, NP242
Past Jeeps: 49 Willys, 81 Scrambler, 88 Comanche
Without "data", all you have is an opinion!