SJ -
You don't say where you are from, but the Silver Comet suggests Alabama or Georgia.
I have a reputation here and elsewhere of being the guy who rides the emptiest roads.
I've done more than 100,000 miles of touring - bad habit - and prefer peace and quiet.
1. Most states have AADT data available through their DOT websites.
Average Annual Daily Traffic is a good measure of how busy the road is.
Traffic is usually busier during morning and evening rush hours and in summer.
But roads to popular parks or lakes can be different - esp. around holidays.
Here's an example of a PDF map of Kansas state highways:
https://www.ksdot.org/Assets/wwwksdotorg/bureaus/burTransPlan/maps/CountMaps/Districts/countmap2018.pdfMany states also have AADT in a database map system such as Indiana & Georgia:
https://indot.ms2soft.com/tcds/tsearch.asp?loc=Indot&modhttps://gdottrafficdata.drakewell.com/publicmultinodemap.aspHere is my general comfort guide to traffic levels:
Less than 500 - Magical
500-999 - Excellent
1000-1999 - O.K., shoulders helpful but not essential
2000-3999 - Fair to poor, shoulders needed
4000 plus - Tough, shoulders definitely needed
2. Next most important aspect is paved shoulders.
A very few states have shoulder width info - although it is unreliable, I find.
Most of us simply use Google Maps Streetview when available.
Quite a few states have Bicycling Maps which are often only marginally useful.
Arizona's bike map was probably done by people who have never been on a bike.
3. Different regions have different attitudes towards cyclists on the road.
Most of us who have toured for years have heard, "Get off the road!"
I've lived in the South and toured all over the U.S. and the South is worst.
(Some may disagree, but look at the number of bike trails for Iowa vs Arkansas.)
Similarly, it pays to use caution around major holidays like July 4th when people drink.
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So, to answer your question -
Close by - there a excellent, nearly empty county roads in south Georgia and south Alabama.
Empty roads, of course, have fewer services, especially with the demise of the country store.
You could use the Georgia AADT info and develop a great tour out of Plains.
Maybe even stop in and see Jimmy and Rosalyn at church on Sunday.
I rode the area 10 years back and can attest to how empty most roads are. Key word - "most".
https://www.crazyguyonabike.com/doc/page/?o=1mr&page_id=72076&v=7YFurther away -
Thousands of miles of empty roads in the Great Plains.
My pick - a loop in the Sandhills of Nebraska out of Valentine.
Even further -
Scenic Wyoming - at the base of the Bighorn Mountains.
Buffalo, Gillette, Wright, Kaycee, Buffalo.
The Hwy 50 section has more traffic, but shoulders.
Which is often the case with road touring - always a section that is busier.
Safe travels - Jama