I'm a mouth-breather (when I sleep) and I wake up with a dry mouth. How can I stop this?

See an ENT. That's a good question. Many individuals breath through their mouths rather than through their nose as a result of allergies, nasal congestion, or from some anatomic blockage of the nasal cavities. The first thing that I would do is see an ENT (ear, nose, and throat specialist for a consultation.
Evaluation. A lot of the time mouth breathing is caused by some sort of airway problem...Enlarged tonsils, deviated septum in your nose, etc. I'd recommend and evaluation by an ear, nose, and throat specialist (ENT) to check for an underlying cause. There's really nothing you can do to prevent the dryness from mouth-breathing while you sleep besides correcting the mouth breathing.
Check with your ENT. See if you have any blockages in your nasal area which is causing you to breathe through your mouth at night.