Is there a verb that describes speaking with a full mouth?
Solution 1:
To talk with one's mouth full. Example: A parent says to a child at the dinner table, "Don't talk with your mouth full."
Solution 2:
Speaking with your mouth full, as a phrase, means speaking with your mouth full of food, but I take it that is not your meaning.
To say something suddenly and without thinking is to blurt.
To speak excitedly and at excessive length is to gabble.
To speak without making sense (due to anxiety, ignorance, or some other reasons) is to babble.