National Parks are quite restrictive when it comes to dogs. Generally dogs are allowed in paved areas, but not on trails. But check the park's web site for details. And leaving the dogs in the trailer is risky if you don't have a reliable AC setup. The alternative is a commercial kennel in a nearby town.
In July both parks will be quite hot.
Several years back when we toured the area in September with a small terrier (16 lb), we found plenty to do by focusing on NF and BLM lands. And state parks. For example Dead Horse Pt SP instead of Canyon Lands or Arches. And Lasale mtns east of Moab.
At that time of year, I think you'd find more to do together by seeking out cooler higher elevations.
