1999 Ford F150 2wd 6 cylinder - What would cause the starter to make a whining sound when you start the truck and will not quit whining and gets very hot on the starter? The Bendix will not disengage.
It sounds like the starter is not disengaging after start-up. This is more than likely caused by a bad solenoid. Once the starter is engaged, the starter keeps spinning because the solenoid does not allow it to release. This, as you discovered, makes the starter get very warm from continuous running. It could also be that the Bendix is allowed to engage, then itself does not retract. This could be caused by a broken or weak return spring in the solenoid, not pushing the core (which is attached to the Bendix) back to its rest position. Either way, I think the problem lies within the solenoid. You may be able to replace this part directly. As it stands, the longer you wait to get this fixed, the more damage will be occurring to your starter motor, which will end up needing replaced sooner as a consequence.
From what I remember 1999 F150's use a starter relay/solenoid that is mounted on the firewall or inner fender. If you follow the positive battery cable it should connect to this relay. If the starter spins as soon as the battery cables are connected to the battery then the relay has welded its self in the closed position and is defective. (This does occur but not frequently, they usually fail in the open position) If the problem occurs after the ignition is switched to the "start" position, you need to determine if the ignition switch is still sending the start signal to the solenoid while it is in the "run" position. On the solenoid/relay you will notice two large wires and a small wire. By removing the small wire you will interrupt start signal. If this makes the starter disengage the ignition switch is defective. (Again this doesn't happen often but can occur) If the starter still spins with the small wire removed then as @Paulster has stated the Bendix return spring is not retracting the drive gear and the starter is defective. This the most common reason but the other scenarios have to be eliminated before you ruin the new starter.