I don't know what the campsite availability will be in late Sept. and I would suggest contacting the Chamber and asking them(or if they have a Tourism Council). We made a last minute decision on May 23rd to go that way from Las Vegas and campgrounds were packed. We ended up staying at Blue Mnt. RV park in Blanding. It is a ways out of Moab but very nice. We can't hike or bike so we were more interested in Four Wheeling and wanted to visit The Needles section of Canyonlands NP. We planned on one or two days there and spent five instead.
The DW met a great volunteer at the visitors center in Monticello, her name is Mary and she know the area and attractions. We traveled up to Moab on Saturday May 28th to see the upper Colorado and were amazed to see the line to get into Arches was 2 miles long and the Hwy Dept. Had blocked the right lane off for the folks trying to get in but had stationed a vehicle at the back of the line saying entry was closed for the day. Last year it supposedly was four miles long and was back into town.
The are other places to see, Muley Point, Moki Dugway, Valley of the Gods, Natural Bridges... Too many to list here.
There are a lot of hiking trails all over the area and the area is unbelievable.
Have a great trip.