Hello lilycamrock and welcome.
My Mom had great success with blue blooming rosemary in Sedona. [White or pink blooming not as hardy]. If you are looking for ground cover, look for low growing varieties at one to two feet high at most.
Arizona cooperative extension agent for Yavapai county says it is very drought tolerant and good for areas down to 10 degrees F. It grows in Mediterranean countries with similar climate to S.W. AZ
