Hi, I have the same problem with Dozer, he has a genetic defect in his jaw, his jaw is much longer than it should be, and he has a severe underbite as well. His top and bottom jaws don't meet together, so his mouth doesn't close, and his tongue is always sticking out, or hanging out, which causes him to mouth breath and drool and his tongue gets very dry.
i don't think that coconut oil would help with the dryness, since it's in the mouth, and she would just eat it or swallow it, and it wouldn't stick to the tongue. I just make sure that there's fresh water available and he can drink when he wants too. You could try putting a product from the drugstore that people would use for dry mouth, something with Glycerin in it, but make sure that it's safe to use on dogs first, by asking either the Pharmacist or the vet. Most over the counter products that are safe for humans are also safe for dogs. Maybe there is some other product that the vet can recommend for dry mouth?