Everything works as should now! It was a bit simpler than I thought. On both sides, there is a harness with red, black, and green wires. There are the same colors coming out of the headlight plug. So I just ran them straight to the plug.
As it turns out, the "over seas" bulbs were my problem! I dug out the original headlights, substituted the Sylvania "made in USA" bulbs, and all is good!
If anything, it was a good excuse to clean up the wires.
Cut out this
Then ran things like this
