I would book it in advance, if at all possible. The Ritz-Carlton has an enormous spa, though it has a very intimate feel to it, as if you are the only guests, but I would still book ahead if you can. And be sure to arrive early.......the Ritz has Women Only areas and Men Only areas that you MUST take advantage of. I didn't tour the Men's area (obviously) but the Women's lounge is spectacular.
If you try the Grand Floridian spa, definitely book in advance. They do have more limited availability.