This is just an off the cuff suggestion for doing this trip. Take the W&OD rail trail from Alexandria out to Purcellville, Va to its end. I would Google map or ask the local DC bike crowd how to get from there to Front Royal, Va which is about 40 miles. In Front Royal you can pick up the Skyline Drive/Blue Ridge Parkway. This will take you much of the way into NC. NC Bike Route 2 coincides with much of the BRP. Bike Route 2 will leave the Parkway at Balsam, NC which is southwest of Waynesville, NC and continue from there west to Murphy, NC. Murphy is in the far southwest corner of NC and is about 130 miles northeast of Atlanta. You would need to do some research to figure that part out as I don't have a suggestion for that.
I'd thought of recommending that approach, but it is very hilly compared to the piedmont or coastal plains. It does have the advantage that you miss a lot of the back-tracking from the Atlantic Coast route. If you do choose the BRP, be sure to get the book, "Bicycling the Blue Ridge" for information on water, food, and lodging or camping.
The approach to Atlanta is going to be an issue. OP might be able to get some help from various cycling clubs and organizations in or near Atlanta.
The Silver Comet trail is similar to the W&OD trail for approaching the big cities. If you want to take that approach (from the west, unfortunately), first find a reasonable route into Chattanooga. I'd suggest taking GA 193/337 to Summerville, then GA 100 down to the south side of Cedartown. Alternately, you could pick up US 27 south of 411 (the traffic between Rome and 411 is intimidating, even in a large car or truck, but 27 is lightly trafficed to the south since they finished I-75). There are 2-3 bike shops on the east end of the Silver Comet, and one of them should be able to help with a route from the end of the trail in Mableton to downtown Atlanta.