We were there in Dec 2011, a couple of times, & we were allowed the "bottomless" milkshake as a drink. We did not get another drink option, except we asked for & got ice water. We did get a dessert in addition to the shake.
By the way, you can order one flavor & the refill can be the same or any other flavor. Even tho after eating the skillet for dinner we didn't have enough room for another shake, my husband wanted to taste another flavor. So, he asked for, & got, less than 1/2 a glass.
Just in case you are interested in the skillet, but can't eat all of it & don't like to waste food, you can advise the server to include only the skillet items you want, rather than all of the items. We get the ribs, pulled pork, corn bread, slaw, & corn on the cob. Sharing that from the 2 person skillet is still a lot, but nothing like the complete skillet.
Enjoy.