Battery Light On
Have you put a volt meter on it with the engine running to insure the system is charging? Tested the battery(new ones can be bad too)?
First i would test the fusible links. There are two, Fusible link A & B. You may want to refer to the wiring diagram below. Next the wiring and connections would be the next things to rule out once the battery and alternator has been eliminated.
Alternator wiring diagram 2013 Ford Taurus