I would refer to it as a "drink" or a "drank"

"Do you want a drink?" "yes" "What kind?" "Fanta"

or

"Yall want a drank?" [see above]

Wait- If they use drank to mean drink, what to they use for the actual "drank" (that is a drinking in the past)?

Interesting. I thought drink (n.) refered to alcoholic beverages!

Where in the US is this?

Alabama.

You are right, its used for alcoholic beverages and non alcoholic beverages so context is key. Ive never encountered a problem with any confusion. Usually when its used, its clear whether alcohol is meant.

Alcohol: "Were going out for drinks" "We have a few drinks with dinner"

Non-Alcohol (Coke/Pop/Soda) "We got some drinks at the gas station on the way over" "Will you get me a drink while you are in the kitchen?" "We have drinks in the fridge"

I think most people that use it would answer "Coke" if asked because thats what southerners are supposed to say, but informally "drink" gets used quite often.