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my standard answer on this is check your grounds. Electrons start looking everywhere for a path to ground when a good one can't be found. All kind of strange things happen. Install toothed star washers on your ground connections, maybe install a designated dash to body ground strap?

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Most of the time when attacking a problem like this, your first tendency is to look for a short in the wiring. Most of the time, it is a missing, corroded or painted over ground point. Shorts are easy to find, high resistance grounds a pain in the lower posterior.
Good luck.
BTW, I would start with the new regulator and make sure it is grounded properly.
Craig