Right indicator electrical short causing headlights to come on

Nothing's shorted. This type of symptom is typical of a bad ground connection. Current going to one light bulb can't get to ground so it looks for a different path, and it finds that through another bulb, then to its mate on the other side, then through IT'S ground connection. In this case, look on the passenger side. I suspect you will see that headlight is not as bright as the left one. That can be misleading too because the low beam will be dim, but the high beam will also be partly lit up. The clue to this is the right headlight will go off if you unplug the left one.

On a 20-year-old vehicle the first suspect should be rust where the ground wires are bolted to the body. Next, look for wires corroded in the terminal.