Dogs and the NPS don't get along well, even if you keep them leashed they have so many restrictions that it makes it impossible to see a lot of the really cool stuff, I don't currently have a dog but several friends do and if the dog(s) are coming we adjust the itinerary accordingly.
I'm another one who can't get enough of the slickrock country. While The Needles District of Canyonlands is off limits there is stuff to explore in Indian Creek before you enter the park.
GSENM is run by the BLM so four legged friends are no problem. Head out to Egypt and spend a couple days taking in the views.
Do you and the dog backpack? Depending on how adventurous you want to be Buckskin Gulch allows dogs, and while in the area you might apply for a permit to hike The Wave thought the May drawing may have just taken place but you can always try for a spot in the lottery for the next day if you are in the Kanab area.
Valley of the Gods would be Dog friendly as would the hike to House on Fire. Not sure about Grand Gulch.
The San Rafael Swell is pretty open and deserving of exploration, especially Sinbad Country.
One thing you might consider is starting south and working north. You might be a bit early for the cactus blooms but could get some here and there.
Easter Weekend in Moab is to be avoided at all costs as are the weeks immediately before and after unless you really like crowds.
Hope the dog can carry his/her own water!
Bob
Edited by Cayuse, 09 February 2016 - 04:16 AM.