Is there anything else you're interested in studying?
I normally would not advise a "game design" degree, because in general I think those programs aren't worth the money; generally the programs seem extremely shallow or overly topical (that is, focusing on trends).
A computer science degree is a good course of action for aspiring game programmers. It's not necessarily a good idea for somebody who isn't interested in computer science (which is not "programming"). Consequently if you're looking at the degree because you think it might be "better" for your career but aren't actually interested in the actual subject matter... I would recommend against it, and instead suggest you take the game design degree program or some other course of study related to things you are actually interested in studying.