Honestly considering your situation I would continue the CS degree and do games on the side as a hobby.
People in the industry like people who can code or are at least a technical even if it's just a bit. I have yet to meet the stereotypical head in the cloud artist in the industry granted I don't have 20 years experience but I did meet a decent amount.
With a CS degree you proove that you can code which is cool, especially for a game designer because then you can talk to the programmers without problems.
On top of it the best way to get early projects (except the ones you might do at school) is to be on students/indie teams. Most of them don't really look for game designers, they are small and they have some ideas that they want to make already BUT most of them need coders . So you can code for them and pitch in some ideas.
A cool thing that I did is that you can judge your level with them too, if you have to apply to all of the crappy teams with no clue about games and get one out of ten jobs or even less you are at the bottom skill level. And then as you improve you start to be contacted by some people, they don't want to pay you but still. And at some point you get interesting teams with a serious project, even if they still don't pay. Etc...
I really used that to judge my level. Just don't work with teams making money and not giving you anything, work for free for teams who don't have any money themselves and are students like you.
Then you'll be at the degree + demo state that frob mentioned and you'll need some luck/contacts/amazing projects to get the first experience (hardest one to get because after you get moved to the first pile) and then that's it you can continue your career.