I wouldn't put sunscreen on a 10 week old, but keep the baby out of the sun with hats, covered stroller or whatever. I took my youngest in October when he was around 12-14 weeks old and we were just fine. He was a very placid, happy baby. I probably wouldn't have done as well taking my first that young, but by the 3rd, it seems easy. It's also really easy if you are still nursing. Naturally, he was off his normal schedule, but he seemed none the worse for it - just slept when he wanted to and ate when he wanted to.
We went to the MK, Epcot and MGM on that trip. We didn't swim at all because we were in a Wilderness Cabin and none of the adults felt like it. The older kids (5 and 3) were OK because we stayed so busy. We did use the car to go to the parks, because it just seemed easier with the stroller. Once I even sent everyone else to Epcot on the Monorail and I walked out to the car and drove over there to meet them. I let my Mom and MIL take the older kids on rides while I sat out with the baby if he was sleeping in the stroller (often.) Other times, one of them sat with him while I did a ride. I have precious pictures of characters holding my son at the Liberty Tree Tavern. We got a lot of extra attention.
Anyway, if yours is a healthy, happy baby, there's no need to stay cooped up in the room all the time. If you want to stay out of the sun, you and your Mom could go with the baby to the tea at the Grand Floridian one afternoon - a nice Mom-daughter place.