Early A switch question

If your dome light comes on with both doors, then yes, the problem is at the headlight switch... or there's a broken wire somewhere. Order a Standard Motor Products DS165 and be done with it. It's like a 10 dollar part.

Sorry, I know next to nothing about Mopar wiper systems, except that there are a lot of them. If you have the schematic for your specific system, I could try to figure out what is supposed to go where.