high resistance in your battery cables and low resistance in your starter from over heating or reversed battery connections at one time or another the combination togethe would create a large arc across the solenoid internal contacts possibly welding them togeter or over heating the solenoid winding resulting in no contact plate movement or starter running out of control
On the Starter
starter is out, motor is locked up or your starter solenoid is out.
It is located in the engine compartment on the passengers side firewall. the solenoid is located under the hood
Starter Ignition Neutral switch
Fusable link? Check at starter solenoid
Usually against the inner fender wall maybe behind the battery
Find the positive battery terminal. Follow the thick cable from that terminal and it will lead to the starter solenoid.
To hookup the solenoid switch on F150 pick-up 5.0 after replacing it, first connect a voltmeter or test light to the solenoid's S terminal. Make sure the starter receives voltage at that terminal when the ignition switch is in the on position.
The negative cable actually is bolted to the engine block, and the positive cable will go to the solenoid that is attached to the radiator support. On newer models, the solenoid would be attached to the starter, and then, of course, the positive cable would connect at the solenoid down below on the starter.
If the starter on your 2003 Ford F-150 keeps engaging after the engine has started, it could be due to a faulty ignition switch that fails to send the proper signals to disengage the starter. Another possibility is a defective starter relay or solenoid that is stuck in the "on" position. Additionally, wiring issues, such as short circuits or poor connections, may also lead to this problem. It's essential to diagnose the issue promptly to avoid damaging the starter or other components.
check your starter, it could either not but getting voltage at all (which would be the starter solenoid) or more likely it is just old or has failed and needs to be replaced
It's located on the side of the starter.