Face down is best for not getting completely soaked. I know it's hard to get water in the mouth/nose this way, but I reckon you'd still get a damp face. Probably easier to learn to cope with washing your face.
If you have a bath with a shower attachment, you can wash your hair by putting your head in the bath and washing it with the shower bit. Or try washing your hair in a sink. How do you wash your face, though?!
I have the same issue i hate getting water on my face as i am scared of not being able to breathe, i usually keep the shower head low so its quite close to me but not close enough to touch my head and lean my head backwards and let the water run that way, that stops all the water from getting on my face