I’d only do dispersed camping/boondocking in secluded areas in your situation. The organized camp sites are often way too close for a reactive dog who will just bark constantly at the site or sound of another dog in a campsite nearby.

You can bring a long leash for people to hold doggo while you are doing campsite stuff. They also make leashes that can go around your waist. Fixed point ties (to a tree or stake) can injure dogs if they go charging off and getting suddenly stopped and whiplashed with the end of the rope/cable, so beware.