You can absolutely find OTs in the hospitals in the DC area, so that's incorrect that OTs are not popular in hospital settings. You definitely have to be persistent, though, in order to get a response sometimes, and depending on the time of year and which hospital it is, they may not have much, if any availability when you call them. The Inova system, which is made up of multiple hospitals, employs OTs at nearly all of them - I did observations at Fairfax (level 1 trauma center and they have OTs in the TICU, among other depts), Mt. Vernon Rehab and Loudoun. Alexandria wanted me to do 100 hours of volunteering before career observation, which I just couldn't do, schedule-wise. Fair Oaks also has OTs, I believe. You just have to be persistent sometimes (just not rude or annoying). Virginia Hospital Center also has OTs (not sure if they take observers), as do some of the hospitals in the city.
There are also a bunch of pediatric clinics in the area - you'd just have to call and see if they take observers. Same with skilled nursing facilities and retirement communities.