[~Zeta~]

Game Design and compute science are very different things. You might want to try the starting courses of each and pick which one you want to do or meet up with the professors/advisers and ask them what they think.

Game Design is in it's name; You will learn to properly design, create, define, plan, etc. to make a fully functional game (you won't be getting to make proper video games through programming for 2 years generally) but it does include programming languages to get you in the right direction. It's more focused on creating the game and its graphics than actually programming the game (this may vary on where you go.)

Computer science is more about creating programs and software through your desired language(s) for processing data, creating applets, etc. It can be used in making games but this degree is not centered around it. You can put up your own effort and integrate it but can be very troublesome.

It's very possible you can double major since they overlap a bit in degree requirements. Not required but it can work out in your favor.


Disclaimer: don't take everything I say as an absolute. I'm providing info for you to decide.