As noted, many NFS CG's shut down in early to mid September. And the second week in October can be iffy certain years. If you can't push your trip earlier by a week, try to plan a loop using SP and NPS CG's.
Heading in from UT, head south from Grand Junction and then east for a couple of days at Black Canyon of the Gunnison. From there, backtrack on US 50 and turn south toward Ouray. Stay at Ridgway SP. Others highly recommend the KOA in Ouray. I know it's a private CG but a great location. Plan a few days in the Ouray area and explore southward to Silverton.
If mountain roads don't scare you, drive south on US 550 toward Durango. The alternative is to head through Telluride on CO 145. Stay at Morefield CG at Mesa Verde NP. The NFS CG's I might recommend all seem to close 9/23. (Hence my recommendation to go a week earlier.) Do take the train from Durango to Silverton. Third week in September is probably the best, but make reservations early as it's a popular way to see the leaves.
From there, head east on US 160 to Great Sand Dunes NP and spend a couple of nights there. Interesting CG.
From there, head north on US 285 to Salida. There are BLM/Colorado Parks CG's open all year along the river in the Arkansas Headwaters Recreation Area. Explore the various canyons heading west from US 24 into thw Sawatch Range. Lots of good leaf peeping opportunities, although by the end of September/early October, many will already be gone.
From there, head north to Rocky Mountain NP. Only limited numbers of campsites will be available by the time you get there, and the leaves will be gone. But so will the crowds. Trail Ridge Road may be closed, depending on the snow situation.
Again, I'd strongly recommend moving your travel dates a week early for more campground availability and better leaf viewing.