I'd head over to Monticello and then south on 191. One of my favorite campgrounds in the world is worth thinking about for a night. And that is Devils Canyon nat forest. The view from the entrance at night is awesome. You can hike to some ruins from the nature trail on loop B. The new loop is open with nice drive throughs but I prefer the originals nice trees.
I first camped here in the 60's, yes and the trees were small then, big now. We knew people and there is an entire valley of ruins close (Montezuma Canyon) that were pristine in the 60's 70's could go right up to and into them. Tells (humps of dirt old ruins) everywhere. My mom pulled a bunch of turquoise beads from the corner of one that was washing out 45 years ago. This area was a center of Anazazi culture.
And, consider the north rim over the south would be a consideration. The South Rim is one big tourist trap imho. Take 163 south whatever you do through Monument valley. Beautiful drive. If you do the North Rim, stay at Jacob Lake that has a full service campground. It is a great basecamp for the North Rim, Zion and Bryce are all day trips with others close.
Just my opinion after fifty years of going to this area.