I downloaded a park map from Shannon's site to research your questions. It looks like the primitive campsites are just south of the Perdernales River next to the Wolf Mtn Trail. I've hiked part of this trail and remember that it's an old road rather than a single track trail. This would facilitate you pulling your cart to your campsite.
As far as safety, you shouldn't have a problem. Carry a whistle with you just in case......it drives those coyotes insane.
