Do you dry camp? If so, we enjoy both Mancos State Park and a US Forest Service campground called Transfer. Mancos SP is easy to find, just look NW of Durango on any road atlas. Transfer, on the other hand, is often incorrectly listed or not listed at all. The road leading to it is gravel, but just take it slow. Here is a link to the Forest Service site that describes it:
http://www.fs.fed.us/r2/sanjuan/recr...transfer.shtml
It's at 8,500 feet so it's quite cold at night and closes for the season on Labor Day. Also, because you are from Missouri the altitude will get to you. Drink more water than you think you need - it really helps. Mancos SP is at 7,800 feet.
We have never camped in Mesa Verde NP, but we did drive through there to check it out and were turned off by the crowds.
If you'd like GPS coordinates of these sites, PM me.
Have fun!