I had this happen on my '51 suburban and I went about 5 years worrying about running out of gas before I finally dug into it.
I had the instrument cluster out as part of a rewire. I inspected the gas gauge and found that the fibre board backing up the contacts had warped and was grounding the gauge. I was able to straighten and reinforce this plate, and the gauge has worked just fine for the past 10 years.
Hey we still need to get together.
Kurt -Burb owner in Raleigh of long standing-