Ford fuel gauges, 1989 and up, use 16 Ohms(Empty) and 158 Ohms(Full)
Ohms is resistance, 16 ohms almost no resistance, like a short, a direct connection
High ohms(158) is like no connection, disconnected wire.
So below Empty would be a short, assuming tank isn't empty, lol
Above Full is a disconnected wire.
Battery would not effect this in most cases.
Some ideas here: fuel gauge over full - Ford Explorer and Ranger Forums "Serious Explorations"®
Read it all the way thru
The wire for the gauge at the tank end is usually Yellow.
If key is on and you Ground the Yellow wire, 0 Ohms, then gauge should go down to empty.
If that happens then gauge and yellow wire are OK most likely problem is the Sending unit(float) in the tank or its Ground wire.
Gauge----(yellow wire)------------sending unit(float)----------Ground