what should I major in if I want to game design experience?
Some schools do offer dedicated game design degrees, so if that's where you want to specialize that can be a good choice. Difference in quality between different schools can be enormous though, so important to research your options well.
Software engineering is also a very good way to get into design, not because it teaches you game design skills specifically, but because it's such a useful skillset for a designer to have.