I'm in the same boat as you. I recommend reading
this website for general information on becoming a game designer.
My guess is that most of the designers from Telltale don't have game design degrees. What usually happens is that people enter the industry through some other discipline (programming, graphics, testing) and then work their way up to a design position. In the meantime, you'll probably have to brush up on your writing skills as well; written communication is really important when it comes to creating design documents and such.