I have seen this problem before and will try to explain the problem and a possible way around it. If you look at the instrument panel at the plug location you will see a hole in the center of the circle of pins. It will have a notch in the edge of the hole. This notch is a keyway to align the plug. Now look at the plug and you will see that the pin (about 1/4 in round) is broken off. Look very carefully and hopefully you will be able to see where the key was located on the missing pin. Mark the plug and socket to show the key location so you can align the plug correctly when you plug them together.
Did that make any sense?

Paul

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84 500 SEL (307,xxx miles)