More than likely, you have high resistance in the circuit somewhere. Either you have a faulty ground, the wiring is corroded, or the socket has corrosion on it.
What I would do is check the resistance on the wiring and then inspect the socket. Lastly, you will need to check the resistance to the ground.
Here are a couple of guides that will help with this:
https://www.2carpros.com/articles/how-to-use-a-voltmeter
https://www.2carpros.com/articles/how-to-check-wiring
https://youtu.be/PxA5wczsCVo
The wiring is getting hot because of the high resistance so I am sure this is the issue we just need to find where it is.
I am attaching the wiring diagram as well so let's run through this info and let me know what questions you have on this.
Thanks
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Saturday, February 19th, 2022 AT 7:09 PM