They have them at Mint Julep, Harbor Galley, and the coffee stand outside Haunted Mansion. They have been automatically giving you the drink separate because they know so many people don't want to have to carry around a sticky cup!
And Pro Tip: if you ask nicely they may give you a water bottle instead of a fountain drink. At Harbor Galley (which also has a much shorter and faster line than the Mint Julep) one CM was just going to give me water in a cup and the other one was like, well water bottles are the same cost as the fountain drink, so gave me one! It was awesome because I could save it until later in the day!