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]
"Do you want a drink?" "yes" "What kind?" "Fanta"
or
"Yall want a drank?" [see above]
Where in the US is this?
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.