Lights dimming sounds like a short in the wiper motor, as the wiper motor otherwise shouldn't be drawing enough power to affect the headlights and there are no other connections between the two systems. Shorts are hard to find - I hate all electrical gremlins like that and I'm not too good at diagnosing that kind of thing. I guess the short could be either in the wiper motor itself or in the stalk off the steering column where the control is. The fact that the mist works is significant, as that uses a different circuit. (I assume your lights don't dim in this case?)
You might try removing the wiper motor and make a visual inspection of the wiring on the outside of it. I had to replace a motor a while back and it wasn't too hard. The motor itself is sealed and I wouldn't suggest breaking into it unless you know what you're doing, but the whole assembly is in a rubber boot and has connections on the outside. Maybe is the problem is simply that the boot has deteriorated and the leads on the motor are touching sheet metal - that would be an easy fix. If not, then you can swap out the wiper motor with a friend, maybe, or get a rebuilt one from autozone (about $50, if I remember right). If you still have the problem, then it must be a short up in the control stalk, at which point you can pull your hair and bang your head against the nearest wall (or swap out the whole steering column, which is about $125 at a junkyard and again not too very difficult to accomplish).