I had a few internal anatomy lessons from Cepheus. The nose and mouth are both connected to the esophagus and the trachea. In theory, I ought to be able to guide water away from one and toward the other, if I'm careful.
We can start with a small amount, and you should be able to cough it up if it goes the wrong way. If you want, I'll catch one of those cats without hair to practice on, first.