Hi
I've just had an upper right molar removed and I have experienced exactly the same drip you describe. The conclusion was that my roots on the tooth they extracted went into my sinus and that by removing it it left a hole which then drained into my mouth until it sealed up again. It must only have been a very small hole in my case as I didn't get any nose involvement, but the drainage was profuse and it was keeping me awake at night. Have you checked the colour of the drainage? Does it look infected? I'm guessing that if one of your tooth roots reaches into the sinuses, (you might like to get your dentis to check) if it's cracked it will flex and release sinus fluid into your mouth.
Bryanna, what do you think? If all that runs true then removing the tooth might not be the only thing that needs doing as you might need a small sinus repair ...again Bryanna?
If your sinuses are infected then that might be the source of face pain too..