Late June is definitely pushing it for RVing in the areas where Saguaro thrive, add monsoons that usually roll in about then which significantly increases the humidity making it the worse time of year to visit. The mountains get rain pretty much every afternoon and with the elevation cools off nicely in the evening, but the valley areas and low deserts rarely get rain, just clouds and humidity for the most part.
The only real positive for going to the Valley of the Sun and southward during the summer season is many of the fabulous resorts that abound the area have their lowest rates of the year. All have marvelous swimming pools and would provide a nice respite from the heat, it is still best to swim early in the day or later in the evening when the sun is less hostile. If you can go a few weeks earlier in June it's not as bad with humidity virtually non-existent.
Our RV only has a single AC unit and the latest we have done any RVing in the lower deserts of AZ is April, with no AC in the rig March would be pushing it. We spend lots of time RVing in AZ during summer months, but all are at high elevations among the ponderosa pine covered in the mountains. There are hotels that are located among the great saguaros, get out early for any hiking or exploring you may be inclined to do.