Oh god, I remember having that same panic, around my grandmother's sister's house. I've got one binder that is stretched out enough just to pull it over my head like I would a T-shirt, but I would really not recommend that for new binders, I tried and thought my shoulders had dislocated. For a long while I had to pull it off how I put it on, get my arms and shoulders out, then fold it down over my stomach and pull down, making sure it doesn't do the rolling thing.
Oh, and congrats on getting the binder!