There are sections with new highway and old highway - but they are only segments.
Otherwise, you are forced out on a busy highway - usually with little to no shoulder.
From Plattsmouth, you can take mostly county roads to Des Moines:
Via Henderson, Griswold, (use caution on Hwy 92) Greenfield, and Winterset.
(There's a nice paved rail trail from Cumming into Des Moines.)
From Des Moines, you can take back roads to Iowa City:
Via Monroe, Lynnville, Montezuma, and Millersburg.
Iowa City to Davenport is a little trickier - more traffic as you head east:
Little longer - via West Branch, Tipton, Maysville Rd, 110th, to Telegraph.
Iowa is very bike friendly in many ways - trails, shops, welcome - -
But its roads are not that bike friendly because there are so few shoulders.
And because most county roads are gravel - paved through roads tend to be collectors.
Still, Iowa drivers are some of the most courteous - esp. on count roads.
State, county, and town parks abound with inexpensive camping.
And the county seats often have lovely B&Bs in old victorian houses.