We have not done route 224, but have used I80 to I76 to I71 to Mansfield OH then route 30 across OH and IN. Route 30 is a good divided highway with not a lot of trucks. We picked up 224 to avoid Fort Wayne and then joined 24, which is divided highway as far as Logansport.
We have also used 34 from south of Omaha to Grand Island then 30 across NE to Ogalalla then 26 on the WY. Good 2 lane with occasional slowdowns for small towns.
We came home from Yellowstone over the recent 4th of July holiday weekend and had no problem getting sites with a call a day or two in advance. Lots of college kids should be back in school at the end of the month so you should be OK.