MS can do whatever it wants, but that does not mean it's attacking your hand. Does this stiffness and soreness ease on its own after a period of time? When you say "very sore", would you describe it as a burning type pain? Is it mainly in the 3rd, 4th, and 5th fingers?

If your answers are yes, you might be experiencing carpel tunnel syndrome. Try Googling it and see if your symptoms match.