I would highly recommend that you go see your primary care doctor
about this issue. There are a few common medical problems that are associated with drooling, and your doctor can examine you looking for these.
In particular, when we sleep, adult humans should normally breath through their nose. Breathing through the mouth, which is the cause of drooling, is almost always a sign that you have a respiratory issue that is causing some obstruction in your throat or nose and that is forcing you to breath with your mouth open.
For example, people who suffer from chronic sinus congestion, nasal polyps, or nasal allergies are often 'mouth breathers' if the degree of obstruction is severe. Your doctor can very easily detect these problems, if they exist, by examining the inside of your nasal passage. If they detect a problem, they may start you on some prescription medications to reduce inflammation or swelling. Alternatively, if the problem is a polyp or other structural abnormality in the nose, they may refer you right away to an ear, nose, and throat specialist
for consideration of a surgical procedure to fix the problem.
Make an appointment to see your doctor at your earliest convenience!