Dead Horse State Park is a must-go if you can get a site. Gooseneck SP is another low traffic area along with near by Valley of the Gods are dog friendly mainly because there is very little traffic. Also near by is Hovenweep NM for some indian ruins. All of the above except Dead Horse are in the bottom SE corner of Utah. Canyon of the Ancients also in the same area. We stayed in Monticello UT at a minimal FHU CG and also in Cortez CO (right across the border) in a very nice CG: La Mesa. It has a lot of trees but no pool ect.
You can drive around Canyonlands with the dogs. And at Capitol Reef you can drive the Burr-Notam Trail with the dogs and explore the slot canyons and spectacular landscape:
LinkLink 2That trail is a backcountry mostly dirt trail but it has a little traffic on it making it safer for in-experienced backcountry drivers. There are even better trails going north from Hwy 24 going to Cathedral Valley but those get you way off the traveled road.
The problem with the big name parks is the dogs; Arches, Bryce, and Zion especially mean leaving the dogs for most of the day if you are going to do any hiking. Zion has a mandatory tour/shuttle bus system.
We like going to Escalante UT and exploring the back roads and slot canyons near there but 4WD is highly desirable. The Hole in the Rock road puts you 70 miles from the nearest paved road at the end for example.
With only a couple weeks you probably should pick one area and stay there. We can, and have spent 2 weeks exploring any one of the big parks and don't feel that we saw everything.