I've been through all those areas many times. As others have said, the later in spring you go, the more stable the weather should be. July typically has lots of rain/thunderstorms which can show up unpredictably. Flash floods are always possible even with blue sky above. Because a large part of the area is open range, (wildlife on the road) I never travel at night.
Gusting winds (crosswinds especially on bridges and passes) ARE NOT your friend, especially towing a trailer.
That said, those areas are stunningly beautiful when the weather is favorable. IMO, Zion, Bryce Canyon, Grand Canyon, Chaco Canyon, Carlsbad Caverns, Petrified Forest, Lake Powell, etc. are must see's but probably not possible in a single 1 month trip.
IMO, daily (sometimes hourly) weather reports ARE your friend.
Chum lee