Funky Fuel Gauge

If the temp gauge works without similar faults, 2 possible issues in fuel gauge circuit alone,
Ground jumper at sender loosing connection due to rust.
Loose gauge. To explain, The stamped tin speedy nuts that support gauge also connect it to the printed circuit board. The contact/mounting studs on these instruments are swedged into a thin piece of fiber board. That and the hollowed out tin nut designed to absorb vibration for 50 years. Times up LOL. Seriously, A lot of owners have replaced the speedy nuts with real hex stock hardware. I wont recommend that. The fiber board on the gauge might shatter like a soda cracker.
One other less likely possibility is a loose contact pin where the wires attach to the board. Loose and/or broken pins most often occurs when the female harness connectors are pulled on.
If this instrument panel has been removed and reinstalled before, the loose pin is more likely.
Bottom line, Sender is always first suspect and inspect simply because its easiest to access.