If you are nervous about being in more public places, choose some of the darker shows. Ellens Energy Adeventure in Epcot is good, as well as The Little Mermaid show in DHS. But of course if your child will be too distracted with all the singing & whatever else is going on, then you can find a quiet bench somewhere.
Whatever you do DO NOT TAKE AN INFANT ON ELLEN's ENERGY ADVENTURE!!!! One of my most horrifying experiences as a mother was when I decided to take my 6 month old daughter on this ride. (It had just starting raining outside, my daughter is usually mesmorized by tv, so I thought it would be a good idea.) I didn't realize that there would be EXTREMELY LOUD noises and scary dinosaurs on it. I even asked a CM if it would be okay to take her and he said, "Sure." -- with no other warning. 2 minutes into the attraction they show the "big bang" with, a BIG BANG, and it set my daughter into hysterics. Once the ride starts, there is no getting off it for 20 minutes. (I know, as my husband, in desperation actually jumped off of the ride at one point when it stayed in the same place and went to an emergency door that actually led outside the park.) I did everything to try and calm her down (including, incidently, nursing -- to get back to the original topic) but nothing worked. At one point, both my husband and I were covering her ears with a sweatshirt we had to try and dampen the loud noise.
I truly thought I had (1) scarred her for life and (2) ruined her hearing. Plus, I felt so bad for the people on the ride who had to deal with a screaming infant (although it was soo loud on the ride, that they may not have heard her that much.)
Not surprisngly, my daughter and her hearing are perfectly fine with no lasting effects from the traumatizing experience --- we still refer to Ellen's Energy Adventure as the "scary dinosaur ride."
Back to the OP's question -- the baby care centers are ideal because they get you out of the heat and in a proper chair, but I would feel comfortable nursing anywhere in WD.
On a side note -- I think, in general, rides at Disney are too loud, especially for young kids. I understand that they want even those that are hard of hearing to be able to enjoy the ride, but I think there would be other options in those few circumstances, i.e., personal amplifiers, etc.