It IS with your sender(s).

Mine behaved exactly the same way, I'd fill the tank and the guage would
continue to register E. The needle would continue to fall, turning the Low
Fuel Warning light, then after a few hours at work, the needle would jump to
the F reading.

The dealership will tell you that you need new Fuel Level Sending units, but
the contacts can be cleaned with relative ease and the sending units will
work again. Mine has been working again for about a year now, and there is
no sign that it will not be working again any time soon. In my case, the
drivers side sending unit had gotten dirty, but there are two sending units,
one on the left side and the other on the right side. Lift out the back seat
cushion, pull back the vapor barrier, and remove the round cover. The
sending units have a connector and a couple of hoses that have to be
removed, then you can CAREFULLY use a punch and a hammer to tap the ring
off. There is a special tool to remove the ring, but I used a 1/4" socket
extension and a small hammer to tap the ring loose, and tight again. When
you put the sending unit(s) back in, take care that you get the gasket in
correctly, it can look like it is OK, but the tank will leak when you fill
it to the very top.

Before you start this project, you need to plan your trips so you are at
home with an empty fuel tank. The obvious admonitions of smoking and sparks
should be observed.