My first cross-country trip was in the fall of 1987 - and I have done many more since then.
You don't mention a few important points:
1. When in the fall - and at what pace?
2. How much touring experience do you have?
3. How often do you plan to stay in motels - which may become more necessary towards the end?
There are excellent monthly temp/precip maps from Prism:http://www.prism.oregonstate.edu/normals/
If you look at October highs and lows, you will see that the West cools much faster than the East.
For that and other reasons, I suggest a West to East routing.
I live in Wyoming and we have already had our first high country snows.
September is lovely, but variable, October can have full-scale blizzards.
Although summer winds tent to be only slightly favorable to eastbound riders, falls winds trend more northwesterly.
That means if you ride west, you will encounter more headwinds. Days of headwinds can be demoralizing.
If you do want to ride E-W, I would suggest heading southwest from Chicago - perhaps via Bike Route 66.
Northern New Mexico and Arizona will be more temperate - and you would end in L.A.
W-E you could do your route in the opposite direction - Starting with the Western Express.
The Western Express is stunning, but a tough ride for an inexperienced rider.
Another, easier, option for a W-E ride is starting in Oregon and riding diagonally.
I might suggest the Lewis & Clark to Missoula, Trans-Am into Wyoming -
Then across Nebraska and Iowa to Chicago.
For any of these rides, I would start as early in September as possible - and it is late August now.
It will probably take, at least, 70 days - putting you on the west coast in early/mid November.
And remember - the days get shorter pretty fast. By November you don't have much riding time.
If you are heading out of DC to get to the west coast - i.e. as transportation -
I would suggest the C&O Trail and ACA to Pittsburgh -
Then you can take the Old Lincoln Highway across Ohio and Indiana.
Unless you need to go into the city, I would take rail trails from Hobart to Joliet.
Then you can ride Bike Route 66 all the way to Barstow California.
If you want to ride up to the Bay Area - I would do Walker Pass on Hwy 178, not Tehachapi.
Then you can ride up the Central Valley either in the Sierra Foothills, the Valley, or the Coast.