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Disconnect the fuel gauge wire at the tank sender. It should simply unplug. With the ignition switched ON the fuel gauge needle should rotate to about the 3:00 o'clock position and stay there. Does it? Now, temporarily ground the fuel gauge wire that was unplugged from the tank. Make sure that you ground the wire to shiny bare metal. With the ignition switched ON the fuel gauge needle should rotate and point directly at empty. Does it?
The fuel gauge wire connecting to the fuel tank sender is 18-gauge tan. If you're fortunate, that tan wire runs down the inside of the right (passenger side) frame rail to a single-wire harness plug located somewhere in the area between the front of the transmission and the transmission rear cross member. If you can find that plug, disconnect it and perform your tests there. The wire you're working with runs forward.
A little sidebar.... Fuel gauge needles bouncing in sync with bumps, turns, acceleration, and braking (in other words, fuel sloshing in the tank) are often a result of dry needle bearings. Fuel gauge needles are mechanically dampened from the factory using viscous fluorocarbon gel, or dampening fluid, that fills the narrow gaps between the needle axles and their support bushings. The dampening fluid gradually dissipates over many years leaving the needles instantly responsive to signal noise as the sender floats bounce around in the tanks. Various solutions are to install a new gauge, have the gauge professionally refurbished, or attempt to recharge the fluid yourself using a syringe and 30,000+ centistoke (cSt) silicone or fluorocarbon gel dampening fluid.
Most wiring under the carriage is covered with (fill in the blank).Use brake cleaner to dissolve gunk to ID the color. Then use the same to blow off the gunk and debris off the connector bd speaks of before disconnecting and testing. Use the same to find a shiny metal surface to conduct your test, although you may have to burnish a location in addition to cleaning. Curious to see how this turns out, I too have a bouncing needle on both tanks.
If getting under the truck is too difficult for you, pop the fuel gauge out and using your meter, measure the resistance at the gauge post. Then compare that to the reading at the cluster connector.