In just a few easy steps we can determine if you have a gauge problem, a wiring problem or a sender problem... You will need a cheap meter - they can be purchase at most auto parts stores, hardware stores, electronic stores or even discount stores.
With the tank empty.
1) Remove the sender wire from the fuel gauge. If it reads full (with the key on) the gauge is faulty. If it goes to empty the gauge is most likely not a problem.
2) Using your meter, check continuity to ground. Since the gauge shows full it will probably show a direct short to ground. Verify this.
3) Drop the tank.
4) Re-verify the sending wire at the gauge to ground with the meter. If it show open, then the tank assembly is shorting to ground (the sending unit stud may be rubbing the body). If it shows a short to ground then there may be a sender problem.
5) Disconnect the sending wire at the sending unit.
6) Re-verify the sending wire at the gauge to ground with the meter. If it shows open now then the tank assembly is shorting to ground. If it shows a short to ground then the sending wire is shorting to ground. You will need to trace the wire to find the short..
7) Using your meter, check continuity of the sending unit to the tank. The sender should show shorted to ground. The sender is stuck in the full position or the sender is faulty.
8) Remove the sender.
9) Using the meter, check continuity of the sending wire stud to the sender mounting flange at the floats lowest level an it's highest level. If the ohms change (from a low continuity to a full short) the sender is working as it should.
To verify correct operation. Connect the sender wire to the gauge and the sender. Ground the sender flange. Move the float and observe the gauge (with the key on). What happens?