I agree with the hostel suggestion. I have stayed at the HI hostels in Philly, Baltimore and D.C. All three were clean but I agree that the hostel in D.C. is a large hostel and it can be noisy. The hostel in Philly is quiet and is a one seat ride from the 30th St. Amtrak station, just a couple of short blocks from the 2nd St. station on the Market-Frankford line. The hostel in Baltimore is quiet but not all that convenient by transit to BAL Penn Station. As someone else said, the hostel in D.C. is 5 blocks or so from a Metro station. Some hostels also have private rooms, but they are a lot more expensive than a bunk in a dorm room. Since you just want to crash for the night I would recommend just getting a bunk in the Philly hostel if you want to go the hostel route. You can also shower in a hostel, which you can't do at the train station. I have used a pay shower in a train station in Europe, and I have heard that train stations in this country used to have them too, but not anymore.
Of course another budget option is to rent a room in someone's apartment on AirBnB. They don't have a filter for good transit access, though, and it can be tedious finding a convenient place on AirBnB.
I do understand not wanting to shell out a lot of $ for a hotel just to sleep in overnight between trains, but when traveling it is important to take care of yourself and get adequate rest. That will be not be so easy surrounded by needy strangers in a brightly-lit train station.
What ever you decide, I wish you a happy trip!
Laura