I would have to see the context in which the sentence was used; but taking a drink is often used as a means to gather one’s thoughts before replying to an important question. If the action is performed by someone who has had too many (alcoholic) drinks, it may be that the words indicate a level of difficulty in performing that seemingly simple task. Perhaps someone better read than I can suggest something more significant.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.