Q: I think my fuel pump stays on and the bat goes dead. 2004 Ford E-150

over the past few weeks my radio and lights on the radio work intermittantly and some times all the power to the radio goes out. the other dyt the bat was dead so i checked it and charged it and the truck started just fine, No radio. then the next day the bat was dead again so i charged it and it started but when i turned the truck off i heard a sound from behind the cab it sounds like the fuel pump stayed on , So i disconnected the bat to shut it off. ..I noticed when i shut the truck off the dash lights falshed on /off a couple times. after key out. Today it started right up i shut it off and a couple hours later it would not turn over just a clicking sound under the hood. i did not hear the pump this time so i disconnected the bat again, and charged it up. No smell of any type I was told it could be a relay but in the 2004/2006 F150 they don't show a relay they say it is built into the fuse block.

My car has 170000 miles.
My car has an automatic transmission.