First guess would be there's something seriously wrong with the wiper motor - it's drawing all sorts of current, causing other systems to fail.

However, if it could draw that much current, shouldn't it pop the fuse? And why would a single sweep (mist setting) work ok, but constant sweeps be bad?

Headlights draw more than wipers - no problems with the lights on, I assume?

This has got to be voltage related - do you have a way to measure voltage while this is occurring? I think Radio Shack makes a meter you can plug into the cigarette lighter. Check for voltage when you switch the wipers on to low speed. At speed on the freeway, you shouldn't see any noticeable drop at all.

Does it do this while sitting still?