Cheap, backyard mechanic test would be to get another gauge (the part you
see) from a junkyard and connect it temporarily (don't bother yet trying to
install it into the dash). If it works properly, you have your answer -- and
your replacement part. If it still doesn't work right, you're looking at the
sending unit (which is *inside* the gas tank -- hence the labor cost for
draining the tank, dropping it along with anything else in the way,
replacing & resealing the new unit, and re-installing everything that was
removed). Good luck (my money's on your problem being the sending unit...)