Welcome to RTA!
My biggest advice would be to get a map of the US -- such as a Rand McNally atlas (get one at a bookstore or at Wal-Mart) -- and see what looks interesting to YOU between DC and Houston. There is so much between the two that would be worthwhile and come under your list of Likes, that I'd be typing here forever. For instance:
Love Music? Nashville. Memphis. New Orleans.
Comfort food? Most of the south has great restaurants with good home-cooking. New Orleans.
History? Natchez Trace Parkway. New Orleans. Battlefield sites. Huntsville AL (if you like space history).
Natural beauty? Shenandoah National Park (Skyline Drive). Blue Ridge Parkway. Great Smoky Mountains National Park.
Grab a map and a bunch of little sticky notes. Start tagging those things you want to see. A route will start to pop up.
As for "cheapest way/best deals", I highly suggest one of two things get coupon books for hotels for the areas you want to stay at, or join one hotel group (Choice, Best Western, or whatever) and start working on rewards. Also find our Saving Money on the Road forum, as there are some really good tips for how to travel on a budget. '
We're here if you have further questions, need for more help, or whatever!
Donna