Short under the dash.
If I turn the key OFF something shorts and the connector under the dash starts smoking. As y’all can see, the connection of the pink wire got pretty hot.
Yes. It's shorting but hasn't blown the fusible link or any fuses. That narrows it already.
Get the wiring diagrams from the factory service manual (not the seperate diagram done by another company).
Write down where each wire and pin goes on the connector. Make a little diagram you can understand.
Disconnect the ign switch connector pictured. Use an ohm meter or continuity light, check both sides for continuity to ground on all suspect terminals.(Assuming the battery is disconnected by now). Then check for continuity to ground on the switch side. If neither turns up anything, try with switch on. Then start testing for cross-wire shorts.
Here's what you know already. It's not likely downstream of a fuse. So its probably before the fuse box, or on a circuit not protected by fuses (alternator field, ignition, maybe horn depending on year/model). But field and ignition should be off when the switch is off. If the short is across wires, that doesn't hold. But ckecking which wires are getting hot will narrow this down.
You also posted that it won't crank. If that means the starter doesn't energize, then it may involve the wire to the starter relay.