If you get in some riding before summer to get your butt and legs in shape, a ride from D.C. to central Ohio is very doable. Most of it (from D.C. to the Ohio border) can be done by riding canal trails and rail trails that avoid both hills and traffic. I'll be taking the following route this fall on a tour from Annapolis to the Mississippi: The C&O Canal trail from D.C. to Cumberland, MD (187 miles). Ride, shuttle or scenic train from Cumberland to Frostburg, PA (16 miles). The Greater Allegheny Passage rail trails from Frostburg to McKeesport, PA (119 miles). The Montour Trail (rail trail)from McKeesport to McDonald, PA (30 miles bypasses Pittsburgh). The Panhandle Trail (rail trail)from McDonald to Weirton, WV on the WV-OH border (18 miles). The remainder of the route to your destination would be on roads; hilly in eastern OH and flattening out as you go west.
The trails not only avoid hills and traffic, the C&O and Allegheny Passage are very scenic. All the trails have websites.