That was our first Ollie trip as well. There's plenty to fill three weeks or more on that route.
We started by making our way to Carlsbad/Guadeloupe. You could start with Big Bend as well if you want. We're hikers, so we spent maybe five nights there and could have spent two more. Be absolutely sure to watch the bats at dawn or dusk at the cave entrance. You could do a quick day trip from there to Roswell if you want to do something hokey. There's BLM land nearby - we camped here the entire time and there are more campsites further down the road, but I'll warn you that the road up the hill to these sites is punishing. It's an oil service road, but we couldn't see nor hear the pumps from our campsite.
From there I think we spent two nights at Cloudcroft. There's a campground there, but it may not be open yet when you go. There's an outdoors store in town that you can ask for locations in the national forest to camp. We camped here. That spot was a bit steep, but the better ones along the road were occupied - just be careful that the south end of that road might be closed and it's a difficult turn around. It's a beautiful area, the town is nice, and if you time your trip right, you can get a tour of the Sunspot Observatory.
From there we made quick stops at White Sands NP and then the Missile Range Museum, on the way to Saguaro. I think we must have stayed a night somewhere before Tuscon, but I can't remember where. In Tuscon, we camped in the forrest up near Mt. Lemmon, which is a great drive. We camped here, and it was a pretty nice spot. You can walk up the hill from there for some nice sunsets.
From Tuscon, we went down to Tumacocori for one night. There's a USFS office on the highway just before town, and they can tell you some spots to camp. We ended up on some BLM pasture land - it was actually pretty nice, but you had to watch where you stepped. If you go there, ask at the park gift shop if they have any home made Membrillo. When we went, one of the rangers was making it from quince that they grow on the grounds. Bring me some. Also stop at the San Xavier Mission if you're into architecture.
Next stop was Organ Pipe NP. When we went, we were the only people there besides the border patrol, so it was pretty nice. Watch out for africanized bees - we were chased down a trail by some. The visitors center will tell you where they are, so don't ignore their warnings like we did. The park has a nice campground and we stayed there. There are some good driving trails in the park but its not much for hiking. The park now comes with less cacti.
Then we went up to Phoenix to see Taliesin, with a brief stop in Ajo where they have a nice town square. I think we stayed at a KOA in Phoenix - I'm sure you could do better with some research.
From Phoenix we went up to Sedona, staying west of town on one of the few roads that allow dispersed camping, here. It's a long, corrugated road, but not too harsh for the trailer. It was a nice, quiet spot.
Then we headed back east to Canyon de Chelly, staying at the campground there. Don't ask Siri for directions there - you can stop at the entrance gate and she'll tell you that you're still three hours away.
May include unpaved roads, lol. They have a warning on their website now to not try to follow Siri's directions. Anyway, Canyon de Chelley is O.K. for a quick visit, but I think the real way to see it is via a tour of the bottom, which we didn't have time to do. Alternatively, you could keep going north to the Grand Canyon, or farther east to Chaco, which is one of my favorites. You could also stop at Petrified Forest NP. Mesa is then just a hop away, etc. So much to see in that area. Another fun thing to see is the Very Large Array, which you might be able to shoehorn in there somewhere. I don't think I'd try getting up into Utah unless you add another week - it's really a separate trip. We've done four 3-week trips so far to the four corners area, still haven't seen everything, and are itching to go back to the places we've already seen.
Finally, whenever we're out there, we spend a few nights in Santa Fe before heading back home, which is sort of a tradition with us.