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To test the gauge, simply have voltage applied to it in a normal fashion and then run a small jumper wire from the "S" terminal to the "G" terminal. If that causes the needle to pin itself, the gauge is okay.
To test the sender, have the sender wires disconnected and connect a ohm meter between the two wires. Now, slowly move the sender's action arm from one extreme to the other. The meter needle should move constant and smoothly from one direction to the other....... NO jerks, dead spots, etc!
Hopefully not a jumper wire but rather a resistor... and if a resistor, yes, it belongs there, it's a needed component.
Reverse the wires of the sender at its connector.
Follow the wires back to its plastic/nylon connector and switch them there. Think about it..... those two wires have got to connect to something somewhere.

Now I'm really confused. I followed the two wires to this wiring block (attached photo). Both wires were connected to the same post