We were out that way in June 2017. Lots to do and see.
When we were at Bryce Canyon we passed the campgrounds and they looked pretty full, and close together. Our original thought was to hike in and spend a night in the backcountry, but that didn't work out. The ranger told us about a dispersed camping area just outside the park. For us it worked out great. We hiked during the day, supper at the campsite, back into the park for a moonlit ranger led hike ( during full moons by lottery) and easy back to the campsite.
It is easier to tell you how to find it as you are leaving Bryce Canyon park. As you leave the park going north on 63, go past the visitor center, then past the park entrance sign and pullout. Just outside the park boundary there is a dirt road going west. I don't remember if there is a sign or not labeling it as dispersed camping but it is in the Dixie NF. There are a couple loops back there, and very few people.
Be sure to take in the sunset while you are in Bryce Canyon.
In Zion, if you hike up the narrows, it is worth it to rent the fashionable water shoes (from an outfitter just outside the park). Sandals are not a good idea in that river. It is pretty crowded at the beginning, but most people don't go too far. And it is beautiful.
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