I'm a proponent of using knobbies where they are called for and have no real problem running them on the road but running them on the road for a few hundred to a few thousand miles before you hit the point where they are needed just wears out the tires and saps your energy. I wouldn't suggest carrying extra tires but purchasing and swapping out to an off-road tire when you get to the point where you need them is easier on you and the tires. No sense wearing down tires just to wear them down.

There are only portions of the Cowboy Trail that are completed but I think you can ride even the parts that aren't finished yet. I was going to suggest it but it is rather far north. On the other hand, the Sand Hills are very pretty and there's a lot of history on the western end at Fort Robinson. There's the old Fort, Toadstool Geologic area and the Hudson-Meng bison kill site in the Crawford area.

Although I am loath to suggest another state, you might also want to consider aiming for South Dakota and the Black Hills and/or Wyoming and Yellowstone rather than Colorado and Utah. Utah gets warm in the summer.