My software shows this as the fastest route:

Leave the DC area by best route for you towards Hagerstown MD. Take I-70 to Hancock, then I-68 to the end, I-79 north back to I-70. Take it to St. Louis, then I-44.

Take the north bypass around Columbus, the south bypass around Indianapolis, and I-270 around St. Louis. Try not to hit any of those cities in rush hour.

Generally, the most affordable places to stay without any hassles are at hotels at the Interstate exits away from any large cities.

If you are willing to drive 8 hours a day not including stops, you can make this drive in 2 days. The halfway point is Richmond IN, which has several reasonably priced hotels to choose from.