No need to stick with one highway number all the way. Just pick and choose at each step of the trip. This is more than 2000 miles.
For example, near home, US20 takes you through Yellowstone. You could then take US16 east through the Big Horns and Black Hills. You might have to take I90 some of the way just because there isn't a good non-interstate nearby. US30 and 26 also take you into Wyoming.
Once past the mountains, choices abound. The main issues are skirting the Great Lakes and the cities along their south shores. US2 is a nice way around the top. US20 tends to go through the cities. US30 is further south, with a lot of small towns.
You'll also find that a lot of the US# highways (and some major state numbers) are now divided with bypasses around towns. Those are a lot safer and faster than the older 2 lane highways. But they are Interstates in every way except name.
And choices that worked well in the Midwest, might not be so good in NY and New England. Getting off the major highways in the hilly parts might not be your cup of tea.