Have you looked at Wes Vann's tech ref section? He has a check-out procedure in there for fuel gages.
1, An open ground to the meter
2, Open sender lead
3, Open sender ground
One of these is most likely the cause.
I work in the dark sometimes on your 60 'cause I don't have much in my material to cover that car. Cars in those days didn't have a separate ground lead to the fuel gage. They relied on the dash ground. Some of the dash grounding schemes weren't the best. The 64 Chevelle relies on a copper strip that goes through the lighter, ignition switch, and gets tied to ground through the original radio mounts. What I did on mine is run a separate ground strap from the instrument cluster, metal in those days, down to the metal dash frame. Stopped a lot of the intermittent dash ground problems.
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