73 Duster Electrical Issue

I cannot tell what colors you have there on the ballast. "One end" of the ballst should have two wires connected together. That should be power to the ballast. If you disconnect all the ballast wires, and turn the key to "run" those two wires should be "hot." I have no idea why hooking th brown to them would cause heating, unless you just left the key on such as:

IN STOCK CONFIGURATION, that is before you modified the car, leaving the key in "run" WILL result in the coil , the ECU, and the ballast drawing current and getting hot.

With the key in run, either running or stopped, that ballast will get hot--too hot to touch