Please do not go to Full Sail. They have very pushy salespeople (based on the experience of a friend’s daughter) but the program is all sales-pitch and not a good education.
What aspect of game design are you interested in? Someone suggested Computer Science, which is an excellent degree that will lead to excellent job prospects both in and outside of gaming. It would be more in the technical and engineering side of gaming. My friend’s daughter wound up in a Digital Animation program at an art school to focus more on the artistic side of things. She had a beautiful portfolio of both traditional and digital art for her applications.