Fuel gauge will read beyond full if the wire is open anywhere from the gauge to the fuel sensor ground. The most likely suspect is the ground at the tank, but you should check the wire all the way back if the ground is good at the tank. There is a ground from the tank to the body underneath. Check there first. Then check wiring from the sensor on back. Also, check the sensor itself with a meter. A good sensor won't read open under any circumsances other than dead. As a wire wound element, they do scratch through and go open.

You can buy an adequate digital meter from Harbor Freight for less than ten bucks. On sale, pick up three for ten bills and have them around the shop. They are more than adequate for making auto repairs.