There's a bunch of great spots in the vicinity of I-70 where it cuts through the reef. There are sites right there between where you get off the freeway (not really an exit) and Black Dragon as well. I'd definitely stay out there over Green River. Some really pretty stuff and great camping. The sites along Temple Mountain road are okay, it's just usually really busy. I'll work up a little map later today when I have a bit more time.
Escalante is a great area for hiking with dogs. Are you just looking for day hikes or longer backpacking routes? Here are a few day hikes:
Sand Creek from the Boulder Mail Trailhead is good and not too long. Dry on the approach but then you have good flowing water for them to play in.
The Escalante River Gorge can be great. The top part coming from the town trailhead is more scenic, IMO but coming up canyon from Highway 12 also has it's perks like some arches, dwellings and rock art if you know where to look. In high water, crossings on the lower section by Hwy 12 could be problematic but it's easy to find out before getting far into the hike. The trail fords the river in the first 100 yards.
Some good choices on the east side of the monument as well. In fact, if I were you, I'd consider connecting The Swell and Escalante by driving through Capitol Reef. Sure, you won't be able to do any good hiking there with the pups, but it's still an incredibly awesome drive to go down the Notom-Bullfrog Road through the Waterpocket Fold and then take the Burr Trail through the reef and into the monument. There's plenty of dog-friendly exploration and good camping in that area. Little Death Hollow and Wolverine Canyons are probably the most dog suitable. They are mostly dry canyons but there are seeps and wet stretches, especially that time of year.
Moving further west from there, The Gulch is a riparian oasis that beckons exploration. The walls aren't narrow or as grand as a lot of canyons nearby, but there is water, shade and plenty to explore.
On to Hole-in-the-Rock. Dogs are allowed in any of the canyons there except the dry fork slots (Peekaboo & Spooky) and Coyote Gulch. Some of the canyons are in the Glen Canyon NRA and leashes are required. On the upper end of the road, you could check out Devil's Garden for a leg stretcher. Harris Wash is good for dogs but not high on my list otherwise. Bighorn Canyon might be good although there is a bit of scrambling down climbing that dogs might not like. Zebra and Tunnel slots are also a decent option.
A really fun, longer day hike would be to go out to the Egypt Trailhead and hike out to The Golden Cathedral in Neon Canyon. You can do it via the b-line route or through Fence Canyon. Definitely check with the visitor center on water levels though. The river can get pretty swift down there in the spring.
Further down the road, Willow Gulch is excellent.
On the Cottonwood Road, be sure to stop and check out the Cottonwood Narrows. Willis Creek is also a must-see and good for dogs. It's on the Skutumpah Road just off of the Cottonwood Road on the north end.
If you have specific questions about any of these suggestions, feel free to ask. I can probably link you to a bunch of trip reports too.