I hope your winter vacation works out. We have been all over the southwest on numerous road trips from our home in southern Calif., and were it not for the Covid restrictions, i would tell you to definitely spend time in NM. In October 2018 we had a fantastic visit there, seeing Taos, Santa Fe, Albuquerque, Gallup and White Sands and Chaco Canyon.
Since that isn't possible right now, Phoenix/Tucson offers a lot, as mentioned by others here. The Desert Botanical Garden in Phoenix is a gem, as is the art scene in Tucson and the Mission south of there. If there is not much snow this winter, I would recommend driving by way of Flagstaff, a charming town that would serve as a good base for day trips to Sedona, Tuzigoot, Montezuma's Castle, the Petrified Forest and the Painted Desert. From there you could head south to Phoenix and Tucson (with a stop in Bisbee on the way). Or you could go straight to the Grand Canyon from Flagstaff. From the Grand Canyon you could stop in at Meteor Crater as you head to California. The early astronauts trained there in preparation for landing on the moon.
Joshua Tree NP in Calif. is truly unique, and there is so much to do and see here, I can only mention a few...Death Valley, the coastal cities, mountain communities, many of which have trails that are often dog-friendly. SoCal's Temecula area has lots of wineries to visit. Palm Springs has canyons to hike and explore, and it is a good winter destination, with a top-notch small aviation museum and good quality art museum. San Diego is chock full water-related activities and a variety of museums, from the bay to Balboa Park to La Jolla, besides being a really pretty city and home to the first of the California missions. (Between San Diego and Santa Barbara there are seven of the 21 missions.) Dog-friendly beaches exist in several cities here, and many hiking trails allow dogs on a leash. You would have to check out the various locations re the hiking trails.
Re. Utah, the beautiful N.P.s there look gorgeous in the winter, especially if the red-toned rock formations are dusted with snow. The colors! If accessible, Zion, Bryce, Arches, Capitol Reef and Canyonlands are worth following the route from Arches to Zion, From there to the Grand Canyon is a good drive. You could go from GC to Flagstaff, do the day trips, then head south to Phoenix and Tucson. Interstate 8 takes about 7 hours to drive from Phoenix to San Diego, if you wanted to finish your trip in SoCal. I won't take the time to tell about all there is to do in SoCal.
These are random thoughts, but 3-4 weeks would give you lots of time to explore and linger. Of course, Covid has upended and restricted access to many of these places, but these destinations are well worth looking into. You could have a truly memorable trip. The southwest/SoCal have SO MUCH to offer!