As mentioned, there are a number of parks less expensive than Cherry Hill. One rarely mentioned is Little Bennett near Clarksburg, MD. You'll want to make reservations because of the holiday anywhere you stay. It's a 15-30 minute drive to the Shady Grove Metro station and parking garage and you can then take the Metro to get into DC. I would advise against driving into the city. Parking can be a nightmare and rush hour(s) driving ain't much fun either. DC does not construct public parking garages.
Someone mentioned Harpers Ferry, WV (also Brunswick, MD). Yes, you can take the MARC commuter train in. Just be aware of its schedule. It has limited trains and hours but if they work for your needs, it's a good idea.
The Mall and the museums are always crowded around holidays. You're starting to get into the time for school trips so there may be school groups at the major museums during the week. My advice is to go in the morning when they open and plan other non-museum activities during the peak time. Don't know how busy the Museum of the Bible is, though my guess is that it is not yet on the radar of many because it is new so it is one I wouldn't worry about being too crowded.
Enjoy your trip