Have you checked your grounds? Is the block grounded to the frame, and the body? Take them off and clean off the paint under the bolt heads. A star washer under the lead will help insure good electrical contact. I run extra ground straps, using the largest cable Oreilly’s had, from the alternator mount to the frame and to the body. So many electrical issues disappeared when I did that, it was amazing.

Is your starter getting enough juice? When you crank, does the voltage go down? Clean all the wire connections from the battery to the starter, if you haven’t already. Can you rig up a voltmeter to the starter while you short across the terminals and see how much it drops? I don’t know if you have starter shims, you might have to space the starter further away from the flywheel, or if they are installed correctly, are the starter bolts tight?
i run 4 gauge from the alternator mount directly to the negative terminals on both batteries in my rockcrawler, and 10 gauge with fusible link from the alternator hot to the battery control terminals and again on to the batteries. All my gremlins disappeared.
You didn’t mention replacing the solenoid, if your model year has a separate solenoid. My 73 F 250 had a solenoid, not a direct piggyback connection. I have no idea what my 14 F250 has, way too complex for me.
It might seem silly, but is there a good electrical connection between the starter and the block? If you live in road salt states, corrosion can find its way into places you can’t even imagine. I’d make sure the block and starter mounting surfaces are clean of corrosion.