You do not say what you prefer, i.e. scenery, mountains, wide open places, points of interest, temps less than 85*, etc. so I am just going with the broadest options. Also, most of the oil & gas operations would be concentrated in North Dakota so avoid the Northern Tier route. There is some in Wyoming but a lot less traffic overall.
If you want easy planning and a wide variety of scenery, culture, etc., pick up the TransAm (TA) in Missoula and take it over to Richmond, VA, where you connect withe the Atlantic Coast (AC). The TA is a very good route, and the maps are very good for the cyclist, i.e. it shows where you can camp, buy groceries, etc. I would personally pick this option if I wanted to not bother to do the whole route on my own, i.e. create your own route.
A slightly more direct would be to exit the TA east of Teton NP and make your way across WY & NE to the Lewis & Clark (L&C) to St. Louis where you could connect to the TA. Not as scenic and you would have to come possibly do some longer days due to services but the truck (and car) traffic would be minimal. Use the WY & NE state bicycle maps (see online) to look at traffic counts, etc. A variation is to start the above but break off the L&C in Sioux City and make your way ovr to Muscatine, IA, and join the NT over to Denver, IN, where you would join the brand new Chicago to NYC route and take it to the AC in Lancaster, PA. This route would be nice in the fall but it is fairly hilly and does have more traffic.
Regardless, you might need to head south fairly soon as it starts to cool off in September fairly rapidly. The plains can still be nice, especially in mid-September to mid-October, depending on the latitude.
Sounds like you are having a great trip, AK to NC!