jamawani 

If you are leaving in mid August to mid Sept - you should DEFINITELY go West to East.
(Since you will have to take a plane or train on one end or the other.)

Why?
The Rockies start cooling off in Sept with early snows at higher elevations -
While the East is still pretty hot in Sept, but lovely in October.
And this year is looking to be an early/cold fall in the West.
It is already snowing in Wyoming and Montana in late July.
(Just a little at 10,000 feet, but it is bizarre.)

Since it can still be blisteringly hot in Nevada and Utah,
I would suggest starting in Oregon/Washington
Rather than doing the Western Express -
Then heading southeast across the country.

If you do a pretty straight shot, you can probably do it in 8 to 10 weeks.
And as for money - the more time you spend on public land - the more free camping.
There's free camping all over the West - but it's rare in the East.
Best Eastern free camping - the C&O Trail from Cumberland to Washington, DC.
(Plus the trail is usually pretty dry by autumn.)

If I were to do a cross-country trip starting in mid to late August -
(And I have done six of them)
I'd start in the San Juan Islands and ride the Northern Tier to Glacier.
Then I cut down to Yellowstone in Wyoming and head east into the Black Hills.
From there I'd cross Nebraska and Iowa -
Maybe swing close enough to Chicago to get a look at Lake Michigan -
Then ride across the Midwest and swing just south of Pittsburgh -
So that I could get on the ATA and C&O Trails.

Best weather. Best chance for tailwinds.
And a lovely time of year, too.
Good luck to y'all.