This is a good one. These guages work off of resistance. The part in the tank has a variable resister, and as the float goes up and down, the resister sends a different amount to the guage. Usually in the form of ground. So, to answer one of your questions, I would ask you one first. Does the needle go to E when you turn off the key. If it does, then probably the sending unit in the tank is bad. Because, the dash part of the guage is getting power and responding to it. It is also getting a full ground which would indicate that the resister is bad, and not producing resistence.
So, I would check your tank sending unit first and go from there. That is not the simple way, but one way or another you are going to have to take something apart. Just run it really low on gas and drop the tank. You might find that the wire is unhooked or has a cut in it. If that is the case than case solved. I would get an OHM meter and check the wire by itself and the sending unit. Then you will know.
Good Luck.
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