You got it. Faulty sending unit. Attach your test light to the positive terminal on the battery. Touching the probe end to the screw on top of the sending unit should show a light (the switch is closed and passes the ground through it). When you fire up the engine, the switch opens and no more ground and dash light (or test light) goes off.
Sometimes you can blow brake kleen, using the straw, up into the switch and bang some crud out and free them up. For what they cost, replace it.