You may have to buck it up and pay the valet parking fee, or park outside the city and take public transportation into the city. I. E., you could take one of the metro lines. From DE, we park in New Carrollton and take the train into the city.
First, it is the height of the tourist season for the area. Second, you will find limited parking the closer you get to the National Mall and attractions. It sounds like your hotel may be a few blocks from the most popular attractions.
Remember, most people will be staying an entire week, so they will probably already be there when you arrive on Tuesday and garages will be filled.
It looks like there are a few parking garages, such as Colonial, but it would most likely be a hike to get to the hotel with your luggage. And you may not be wanting to walk several blocks in DC that late at night. Just a precaution.
Also, I will warn you, living in DE near the resort beaches, many, many people own beach houses in DE from DC. They start leaving mid-afternoon Friday night up until early evening. The later you can leave, the better off you are, especially if you are heading towards the Bay Bridge to get home.