From my experience this time of year in those states, you should be fine with the campgrounds, though weekends may still be crowded at some. As you noted, pay attention to highway closures and detours. Each state DOT lists their highway work on-line, and it pays to check ahead. Lots of highway construction going on this year, and it will until the weather brings an end to the season.
It sounds like a wonderful adventure, and the pace you mentioned sounds perfect! Good advice on the spare fuel, especially in the mountainous areas. The peace of mind is well worth the effort of carrying a gas can around.
Have a safe and enjoyable journey!
2016 Coachman Leprechaun, 319DS, Motorhome, Dodge Durango Toad