Is it possible for you to drown yourself with saliva?

Yes: If someone have excessive saliva and can't swallow their own saliva they can have difficulty in breathing or choking situation.

Asphyxiation: Yes. It usually happens only in those with no cough reflex or unconscious as they don't have the ability to cough up nor have the stimulation to cough such as in those above or those under sedation or narcotics which depress the cough reflex.