I don't know of any butcher shops in Alpine, but there is a supermarket off the main highway that has dry ice. It is at the intersection of the road to head into the Grey's River.
Hog Island had their large trailer set up and were busy as hell when I stopped by there last Oct. after an elk hunt. They used to have a nice shop on the south side of Jackson, but that it no longer open. The trailer is right by Daniel Junction.
The meat locker in Jackson will most likely not hang your game if you aren't having them process it... they are overloaded with animals usually with hog island no longer operating south of town.. so they only make room for animals they are being paid to do