The sending unit is the ground. You can look up the resistance value of the tank at 1/2 full. Disconnect the sender and use a resistor, or pot of the proper value to ground and go for a drive,you should get a steady 1/2 tank reading. If you do not get a steady 1/2 tank reading it's a wire or the gauge itself. If you get a steady reading it's the sender. 0 ohms empty 90 ohms full so a 45 ohm load will tell the story.
I don’t think the sending unit is the ground, there should be a ground wire that runs from the sending unit to the frame. The tank just sitting in the saddles would be a terrible body ground. It actually sites on rubber or tar paper in there from the factory to prevent squeaks.
If it is bouncing from full to zero constantly, sounds like a ground wire problem. You could try to run a ground wire from the frame to one of the flaps of pressed welded tin on the tank with a self-tapper and see if it fixes it. Of course if there is a lot of corrosion around the sending unit where it meets the tank, this won’t work, but it’s easier than pulling the bed off as a first step and the sending unit is under the cab anyway, not the bed so the way to do it is drop the tank.