All Smithsonian museums are free. There are few museums in the area that are not part of the Smithsonian, and will cost. One of them it really worth seeing - The Newseum, which about news. Fascinating - a lot of artifacts of famous news stories. The ticket will cover two days.
If you like Modern Art - the Phillips Collection is also privately run, but worth seeing.
If you want to go to restricted government facilities, contact your Congressional Reps office. They have tickets for their constituents.
If you are staying at Cherry Hill, you should also consider a day trip to Annapolis. Baltimore also has some interesting areas.