I wouldn't stress - I think you'll be fine, especially if you are in the parks during the weekdays. The holiday weekend and most of the fall breaks are over. That was the biggest thing - folks tend to paln their time off around that weekend. I think we will see significantly lighter crowds for the next week or so.
I used to work for the VA Hospitals, so I understand where you are coming from. Remember, there are actually a lot of fairly quiet "chill out" spaces in both parks.
DL spots:
The upstairs of the Hungry Bear is really peaceful. The nook by Baby Care and First Aid is nice. Big Thunder Ranch is good. The area next to Small World (where the parade comes out from Backstage) is extremely empty, as well as the overflow queue area for it. Upstairs in Innoventions is good. The riverboat is a nice crowd break. Princess Fantasy Faire has some quieter areas.
DCA Spots:
In the back of the Hollywood Studios area by Monster's Inc and Muppets. Lots of quiet open space there. Golden Dreams area - the Amphitheater by the Lagoon. The Cove Bar above the Ariel's Grotto Restaurant. The Wilderness Play Area. The path that goes to the right through the Grizzly Rapids attraction - really quiet and nice music. The benches in front of the now closed Sun Wheel - overlooking the lagoon. The Bountiful Farmers Market area.