hi I am quite embarrassed to admit this but I too dribble when I am asleep and have done ever since I can remember! It is because I cannot breathe through my nose so therefore sleep with my mouth wide open I even wake up and realise it myself! My daughter must also be the same because I have noticed marks on her pillow the same. I don't really no any cures as such but if I don't have a cold I use nose spray ie otrivine/vicks etc enables me to breathe through my nose most of the night and this definitely helps.