First, I wouldn't head to Colorado Springs. So many better choices in CO. And while Moab has great scenery, it gets pretty hot (average near 100), esp. later in June. And there aren't trees so there's precious little shade. It does cool off nicely in the evening.
I am assuming there are kids involved if there are "Six of Us!". If so, Moab (and Arches NP specifically) is great for kids. There are enough hikes and things to do in Arches, you may not get up to Canyonlands. That may be just as well, since Canyonlands means more driving while Arches means climbing on and around arches and playing in the sand. Hike out to Double O Arch and take the primitive loop back. The Windows Section has a lot of short hikes. Fiery Furnace is an interesting guided hike for kids. Head out to Delicate Arch to watch the sunset. If it's warm, take a raft trip one day while you're there. If you don't have A/C on your LG Silver Shadow, try to reserve a site in Arches at the Devils Garden CG. Incredibly fun CG for kids. Sometimes there are deer in the CG in the evening, plus plenty of rocks to play in and explore around. Many arches can be reached on trails right from the CG. But reserve ASAP as it fills quickly. And be prepared for heat.
If that just seems too hot, then head north from Cortez on CO 145 toward Telluride. Gorgeous mountain scenery all along this route. There are several NFS CG's up near Lizard Head Pass (Matterhorn, Cayton, Sunshine). Reserve your first night there. For the second night, head up to Black Canyon of the Gunnison NP. Great campground and one of the darkest places in the U.S. Great stargazing. Higher up so cooler. Take a drive down the East Portal Road to get to the river and look up from the bottom of the canyon.
One can get to Dinosaur NM from Moab either by taking US 191 or heading back into Colorado and taking CO 139 north from Loma, which is the road to take if you stay at Black Canyon. Plan on leaving Moab or Black Canyon early so you have plenty of time in the Vernal area. Make sure you get the kids to the "bone wall" aka the quarry at the NM. Also, in Vernal don't miss the Utah Field House. Lots more dinosaur stuff and a great combination with the NM itself. The kids will love both.
That evening, try to camp up by Flaming Gorge NRA. Quite a bit higher up there and a welcome respite after a few days in the hot desert. A number of NFS CG's up there are reservable.
What ever you decide, get going on the reservations.