The shortest route is 2249 miles, so if you want to do this in 21 days, you don't have much wiggle room. The support vehicle, however, gives you some options if you're not too picky about riding every mile.
The shortest route, however, is mostly interstate, so I don't recommend it (and bicycles are prohibited on much of it anyway). If you ask Google maps for driving directions with the "avoid highways" option, you get a route that is only 113 miles farther than the shortest route, so that's not too bad. Still a difficult ride for three weeks.
A more pleasant route would have you take the ACA Southern Tier route to St Augustine and the ACA Atlantic Coast route up to Richmond. This would be 4183 miles, however. I would recommend it nevertheless. Perhaps you could ride in your support vehicle for part of the way, or perhaps you could find more a few more weeks time. If you're going in the spring, it is probably advisable to stay southerly to avoid possible late winter storms. The Rocky Mountains in Colorado are not reliably passable by bicycle in early and mid spring.