Whew! That's quite a trip you've got laid out!
As for maps, I'd suggest you contact the various state highway departments and ask them for state maps to begin with. Note that their maps typically will only show larger roads, which may not (will not) be ideal for cycling.
Next, I'd suggest you buy a copy of something like DeLorme Topo to look up the smaller roads. Note getting it onto a netbook, if you decide to use one on tour, will require you to copy the DVD onto USB thumbdrives.
From Shiloh to Chattanooga, I'd suggest looking at U.S. 64. It's a major highway across southern Tennessee, so use Topo to look for parallel county and state roads. When you get to the top of Monteagle Mountain, jump onto U.S. 41 down to Jasper, and on into Chattanooga. Getting through Chattanooga to the east could be tough, but you'll want to try to hit Chickamauga battlefield. There are some ways to get out of town going north, but the last 30 years have seen a lot of development to the east of Chattanooga. You might try crossing the Tennessee River around Dayton or Spring City, and going over towards Madisonville.
If I were going to try this, I'd want to get out toward U.S. 411 up through eastern Tennessee, and maybe try going up 321 from Townsend/Pigeon Forge. (Pigeon Forge and Gatlinburg are heavily trafficed by tourists, though.) Once you get to Elizabethton, go up through Shady Valley to Damascus, Virginia. I'd suggest taking the last 1.5 Adventure Cycling Trans America maps toward Roanoke and up toward the Shenandoah Valley. Personally, I'd go up the Shenandoah to Antietam and Gettysburg, and then probably back-track down to about Waynesboro, and cut over toward Richmond on the TransAm. Manassas has grown too much in population, and not in road infrastructure, for my personal taste.