Testing procedure I used in my '55 Courier wagon:
Remove guage and sending unit from the car
Set them on a table near battery with alligator clip wiring sets that will reach the battery. Connect ground from batt to guage case bracket and jumper over to sending unit case
Use other test wiring to connect positive battery lead (12v) to guage.
Run a test wire from guage to sending unit terminal
The above wiring mimics car wiring.
Set the sending unit up on two wood blocks and slowly move the float up and down and watch the guage needle while doing this.
If needle movement corresponds to float movement, your instruments are OKAY
It may not mean your guage is perfectly calibrated with the sending unit. hat is another matter.
If instruments are okay, check your wiring in the car with an ohmmeter. There should be maximum resistance when testing each positive lead with the negative lead touching good body ground.
Chances are good that the long wire running back to the sending unit could be damaged
Is that ancient condenser still connected at the sending unit terminal? It may be a bad condenser.
I just removed my condenser entirely, but you could replace it with a ignition condenser if you want the guage needle to move slowly as gas jostles around in the tank during starts and stops.