Oooh, good question. Both could be correct, but the emphases are slightly different. "I've got to get changing" is uncommon and puts the emphasis on the action itself ("changing," probably changing clothes). It's similar to, "I've got to get going" or "I've got to get moving": they all emphasize whatever action the gerund (the -ing word) is. Normally you see this grammatical construction with big physical movements or changes in position, so it's a little unusual to use it with "changing." To my mind, "I've got to get changed" puts more emphasis on what your state is AFTER the action happens: Right now I am not changed. I've got to get changed. In the future, I will be changed. The emphasis is on the change in state.

If you are talking about changing clothes, then sentence A is correct.