Jace,
Those senders operate on resistance (an arm wipes across a rheostat causing voltage changes which goes to the gauge). It sounds possibly like a bad sender caused by ware or corrosion (remove and replace). It could also be caused by a bad float (they can get old and leak like other things in life
). The float is in the tank and the fuel level determines the position ofthe wiper in the sender. I believe that the sender should be accessable from inside the trunk and should be available possibly from a local parts store or for sure from a company such as Dearborn Classics, OEMFord, Obsolete Ford, just search for Fairlane parts and I'm sure you will find links and parts.
good luck,
Mike "monster"