To sum up what others have said, you will find a number of NFS CG's closed soon after Labor Day. The few NFS CG's left open get busy as there are far fewer choices for all of us who are still camping in Sept. Plenty of boondocking still available at that time though. At some of the more popular NP's though, you might have problems finding a spot. Places like Zion and Arches are already getting pretty full and will probably sell out for certain dates as we get closer to fall. I'd get on recreation.gov to research the CG's you are interested in and how many sites are available as time goes on. You will also be limited because of trailer length in some CG's, as they were built when many people used tents and few had large rigs.
With only two weeks for your trip you will be severely limited as to what you will be able to see. While you may be loathe to make reservations, you might want to do a lot of research before you leave to decide what you want to see. When towing, you will also find that you won't make great time on many of the roads through CO. I think you're going to find this a rather rushed trip if you're planning on sightseeing in CO, UT and ending in AZ.