Fuel pump issues.

Hello, have you tried turning off the inertia switch for the fuel pump circuit and see if that shuts the pump off? That would help to rule out if the pump is getting power from another circuit it might be shorted to. The inertia switch on these is down on the passenger side kick panel. Its pretty easy to find, its just a 2 wire switch, looks like a button with two wires plug into the bottom, a Pink/Black wire and a DarkGreen/Yellow wire, they are 14 or 12 gauge, pretty heavy wire. But you can just unplug the switch and it should stop power from going to the pump. And if the pump continues to run, then you know its getting power from another source. The wiring harnesses in these trucks is pretty basic, we have an 01 as well for a spare vehicle, they are pretty reliable.
So pulling the relay didnt stop the pump from running? Is that what you mean by it was Ok?

No power runs through the switch to the pump, the inertia switch is a safety component. If you are in an accident and hit hard enough, the switch will open and stop the fuel pump from running. Disconnecting the switch will help to determine if the power is coming from the relay staying on, or if power is coming from a short to power somewhere else on the circuit. Ill post a wiring diagram of it for you. But even pulling the relay out, if the pumps stops, then you know either the relay is staying latched (On) or something is keeping the relay on, as providing a ground for it to stay active.

Nope, lets start with the relay, if you pull the relay and the pump shuts off, then either the relay is staying latched when it is deactivated, (this happens when there is some carbon build up on the relay contacts internally), Or the relay control wire (which is a ground the PCM controls, Ill post wiring diagrams of that as well) is staying grounded by the PCM or that wire itself is shorted to ground somewhere. This is pretty easy to check with the relay out, you'll just need to check the control pin where the relay sits for continuity to ground with a multimeter.

No worries, ask as many questions as you need to, thats why we're here. The truck doesnt need to be running, if you hear the fuel pump staying on after the key is turned off, just start with pulling the relay, and we'll go from there.
We can even start to track down where the power is coming from by checking for voltage at certain locations.
This first diagram is battery power coming from the fuse through the relay and out to the pump.
The 2nd diagram is still that same DarkGreen/Yellow wire from the relay, you can see it runs to a Splice (S111) where power goes two ways, to the inertia switch and then the pump, and also power goes to the PCM where it monitors whether the relay is actually suppling power to that wire. You can see at the inertia switch the wires change colors, DarkGreen/Yellow going in and Pink/Black coming out.
The 3rd diagram shows that PCM monitoring wire.

Im assuming the fuel pump is staying on all the time?

Well technically the fuel pump should only come on for about 2 seconds when the key is first turned on, this is to prime the pump and get the fuel pressure up to spec. The PCM will then wait until it sees an RPM signal to turn the fuel pump relay back on again and allow the engine to start.
Are you having an issue with starting at some points?

This can be a bad relay, even if it looks ok or the fuel pump has a bad spot on the armature.
Have you tried a different relay yet? On these trucks there are a few relays that are the same in the engine compartment fuse box. I am pretty sure the AC Clutch relay or Trailer Tow Relay is the same as the fuel pump relay, so you can swap that and see if the issue continues, if not then replace the relay, if it continues to be an issue with starting, then we will need to diagnose the fuel pump and its wiring.
Do you have a multimeter and test light to do some testing with?

You can check for power right at the inertia switch when the truck wont start.
So you are not hearing the fuel pump when the starting fault is happening? It is very common on this trucks, you have pretty high mileage at this point.

Go and find the inertia switch first so you know where its located. That way when the fault occurs you can be ready and go right to the switch to check for power.

Did you replace the fuel filter as well?

Ok thats good. Has the check engine light been on at all lately? The pump staying running and the starting issue sounds more like a wiring issue, since we're kind dealing with two potential faults here. It would be good to check fuel pressure when this is happening, but this will get into more technical type testing if youre comfortable with that. You will need to pick up a fuel pressure gauge, harbor freight has a cheap kit you can get and Ford has a very easy port on the fuel rail where you can screw on the adapter. If you look around the fuel rail where the injectors are, you will notice a small cap screwed onto a threaded port. Its difficult to miss. Ford makes this testing pretty easy compared to other makes.