Similar issue here too, '00 E250 5.4 gasser motor with original fuel pump and sender, 275K + miles!
At about an honest 1/4 tank and a few miles later needle drops to below E, no other symptoms or engines issues. Filling tank doesn't "cure" this until a bump or rough road seemingly jars the float back to life, suddenly all is working perfectly once again.
Following an EVTM schematic tracking the fuel pump's wiring harness connector ground would be easy enough. "Flexing" the chassis wiring connector while the meter is in continuity mode might reveal intermittent short--the task then tracing where along that wire run the damaged conductor has failed.
Mostly though this sounds like the sending unit inside the tank hangs up momentarily. If the wring to the pump is sound with no shorts the only way to cure this is replacing the pump and sending unit---requires dropping the tank.
Something that might work but would take a while to be noticeable is using Techron Concentrate as recommended: 20-24 oz added to a full tank (22-25 gals) every 3K miles or so. Even though this is a good all around practice for fuel system health it could positively affect your sending unit issue.