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[oser1ww]

Hello everyone here, this is my first post after a decent amount of lurking.

I have a problem with my car, it keeps shutting off randomly, especially under load. I did make sure to read older topics and try them out, but to no avail. 
My car is a 198X (post 1984) Mk4 998 fitted with a HS4 carburetor and has an aftermarket A/C put into it, though I am unsure how long ago was it installed.
I have noticed that the wiring from the ignition switch (brown, red/white, and the auxiliary green) also becomes extremely hot when the engine is running. So much so that it did smolder once under the dash during a test. After that I have decided not to use the car until the wiring issue is fixed. 

The wires are equally as hot when the headlights/ aircon are on, as with only the engine running. Regretfully, I have not yet tested it alone with the ignition in the No. 1 or 2 position though. The wiring is in some areas a bit dirty, but the problem persisted even after cleaning with contact cleaner. 
I did not that the battery has a failed cell in it (reading only 10.5, rather than the accepted 12.6V with the engine off, reads slightly above 12.7 when charging). 

Thank you for reading.