No, a gauge that pegs to full might be fine. That behavior could be a result of a broken lead somewhere in the gauge circuit.
Here are some causes/remedies that are described in Shop Manuals. Disconnect the lead from the gauge to the sender - a good gauge will peg Full. Then, run a lead from the sender post on the gauge to a good ground - a good gauge will read Empty.
After that, test the sender unit (outside the tank). These are simple to "bench test".
