Electrical issues.

cudamark said

But, this is a '64 we're working on.

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Refering to my comment to Dan? My point was that 'very rare' doesn't mean its correct or incorrect for the OP's particular car. Its just a generalization.

vpmopar said

I have been out working on this problem with your suggestions and drawings. Looking for a disconnected ground that I may have missed. I will continue to look until i get it right. Thanks for everyones comments. I will update as soon as I have some thing to report.

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Here's a sketch that includes high beam indicator. I don't know if the connection locations or colors are the same as on your car.

After Dan's comments and thinking further, I really think its worth checking if the issue shows up with the switch in Park as well as in Headlights.
Another thing to do is to take out lamps one at a time.