We were there last week, too. The server told us right away that the shake could be ordered as a drink, but it was not refillable. She did bring glasses of water to all the shake drinkers, though! They also got dessert. BTW, this was also the case at 50's Prime Time--you could order a shake and a dessert (and my DD loved the PB&J shake there!). Have fun--it is great.
Sheryl