You need to take a continuity tester to the sender when you have it out of the car. I'm assuming you disassembled it when you had it out? It's a relatively simple job to do.
The reserve light works on the principle of the float reaching a point at the bottom of its travel where two pins touch a metal disk, completing the circuit. Something is shorted together in this circuit, causing the light to be on at all times. With the sender disassembled you can see the guide wires, float and disk clearly, so if there is something that is causing them to short out or touch at all times it should be obvious.
With the sender removed from the car, unscrew the little nylon or plastic nut on the bottom. The metal disc and a plastic disc will come out. Pay attention to the orientation of the plastic disc, as it has a series of slots like a labyrinth to dampen the movement of fuel in and out of the sender body. This is what keeps the float from moving up and down rapidly while driving. The slotted side has to be against the metal disc to seal the labyrinth, as it acts as e “roof”, so to speak.
With the discs removed, you can carefully slide the outer tube off of the assembly, exposing a metal rod that the float slides up and down on, along with some fine copper guide wires on either side of the float.
If you look closely, you’ll see there are copper “fingers” that project out from the sides of the float that make contact with the guide wires. The guide wires are nichrome, and provide the resistance to make your gauge read the fuel level as the float moves up and down. The fingers have the nichrome wire threaded between them, but these can become dislodged if the sender is damaged to handled roughly. Make sure the float moves freely up and down the full travel of the rod. If it binds or hangs up close to one end, look at the fingers and nichrome wires, as this indicates the fingers aren’t on the wire properly.
On the bottom of the float are two pins. You need to make sure these are clean. If you connect a test light or continuity tester to the terminals for the reserve light and move the float to see if the circuit opens and closes as the float contacts the disc.
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Dan