Sounds like it don't it?
First thing I would try is to ground the wire for it at the sender and see if the gauge stays stable.
If it does I'd go after the sender.
If not I'd be looking for a loose connection or a bad spot in the wire.
Not finding a problem there ground the wire at the cluster.
Still wild, cluster tracer connection or plug connector.
Not that then the gauge itself.
All the other stuff is free just for the time, but the gauge usually not so much.
