I don't know what year your car is, but on 65's and 66 cars with original wiring, some of the wires in the harness going into both of the tail light housings can tend to crack, fray and seperate causing operating and shorting problems simular to what you are describing - I have repaired a few of these over the years. Its worth taking a quick a look.
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1965 K code coupe - numbers matching - restoration ongoing. 1987 LX notch, stock heads and intake, 3.55's and typical bolt on's 13.89@100.25
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