How do I mitigate moisture problems when putting up drywall over stucco with torn tar paper?
Absolutely not.  Drywall in a moist area is a sure fire way to get mold growth - you have to address the water issue first.

Since the walls have no wood sheeting, you could use a closed cell spray foam insulation inside the walls applied directly to the back of the stucco.  That will block water from getting in and act as a vapor barrier as well - then you can drywall over that.

If you had wood sheeting on the walls with tar paper and stucco over top then I'd be concerned about moisture getting trapped between the tar paper and the foam.