cigarette lighter wiring

probably have a bad socket or most likely a bad element shorting it out. look inside the socket and make sure nothing is in there - a penny, paperclip ect. but yeah unplugging it will not effect anything else.

the other lighting problems are likely not related as all back lights would be out and not just the left side. for that check grounds, but in the 67 they have a tendency to corrode in the rear harness connector in the trunk on the left side.