Mesa Verde has a large campground. If you have a Senior Pass it would be 1/2 price - dry camping so it would be very reasonable.
We've stayed at South Mineral Creek many times - lovely mountain view along the stream. July is the busiest time there. Hopefully, someone who's there now can help answer your question.
If you've done Silverton to Ouray in your Jeep you know what the highway is like. So only you know if you should do it with your RV. We do it all the time with our 40' motorhome and love the drive but everyone is different. You won't have any tree issues, for sure.
Have fun Jeepin'!! It's one of the very best places in the country.
We wish we were joining you!