I have AT&T, and I know I at least had cell service in Skagway because I specifically remember sending some picture texts there. I can't remember if I used it in any other ports. (I was on NCL, not Disney, but that doesn't really make a difference.) I did turn off roaming so that I wouldn't get any service unless it was an AT&T signal, and I didn't get any extra charges.
ETA: I am not sure if you were talking about on the ship or in ports. I was too scared to try it on the ship, so I am no help there! It's such a port intensive cruise, though, that I would just wait and use it there and enjoy peace and quiet on your sea days.