Well, every park has a Baby Care center. You could check that out.
At Epcot, there are lots of quiet, cool corners in Innoventions. I can't think of any convenient places in the other parks (AK will be tough) but maybe others can.
Edited to add: At MK you could take the monorail to the Poly, which has a great lobby to sit & relax in. At the Studios you could do the same thing and head to the Boardwalk or Yacht/ Beach Club, which have large lobbies with sofas to relax on.
There are various indoor CS restaurants like Cosmic Ray's or the ABC Commissary, but they are generally very loud and somewhat chaotic. I guess upstairs at the Columbia Harbor house (MK) is one possibility, if it's not crowded.