You should not have any trouble in September by just showing up. That is what we have always done.
Edit to add that I have just made reservations at Spruce Lake in Estes Park. There you DO need reservations in September.
Another fine campground is in Creede, CO: Mountain View/Riverside (name from memory.) It is one of the nicest in Southwestern Colorado.
We also have reservations at Cheyenne Mountain State Park near Colorado Springs. We were lucky to get them.