I have really debated with myself about contributing to this thread again or not. To me, the answer and diagnostics needed to figure out the issues have been posted multiple times.
Here are some facts:
A fuseable link wire is made to work like a regular wire and then burn up if it gets to a specific temperature so that the rest of the wires in the circuit will not be damaged. A fuseable link that is bad will not conduct electricity. So, if something is on continuously then a burned fuseable link is not the cause.
The small purple wire at the starter solenoid delivers 24 volts to the solenoid and turns the solenoid on which turns on the starter motor itself. That wire goes from the starter motor to the through the firewall plug on the drivers side of the firewall behind the fuse panel. Using a 10 mm socket to unbolt that plug will let it come apart so you can look at the connections. There could be arcing between circuits inside that plug. The only way to know is to take it apart.
From that plug the purple wire goes to the starter relay above the throttle pedal and below where a radio would be in a civilian truck. That is the entire path of the purple wire.
A starter that turns itself on with only the big red wire attached to the solenoid has a failed on solenoid. Or, the big red wire is attached in such a way that it is conducting with the S terminal. This can be tested with jumper cables attached to a starter sitting on the ground.
The starter relay can fail. Stock or any aftermarket relay.
With the purple wire removed from the starter motor solenoid and the batteries attached. Nothing should happen. If the starter turns on, see 3 lines up.
With the purple wire hanging free inside the truck from the starter relay and attached at the starter motor solenoid. Nothing should happen. If the starter turns on after it has been tested with the purple wire not attached at the starter solenoid. Then the purple wire is getting voltage from somewhere it shouldn’t. The firewall plug is the most likely place for this. Otherwise the purple wire is going to have to be followed from the starter, around the back of the engine and over to the firewall plug.
Starter relay, starter wire (purple wire) or starter solenoid are really the only things that can be wrong.
Here is a video about why there is a starter relay: