We've been to all the mentioned areas - in the right seasons. :)
I think southern Utah in December would even be too cold and you'd still have the possibility of not seeing snow - just wind and cold. Winter driving with an RV in snow country can be dangerous and some areas require you to put on chains on the RV.
I would highly recommend staying at Dead Horse Ranch State Park in Cottonwood, Arizona. Taking a day trip to nearby Sedona or a little further to Flagstaff and the Grand Canyon would almost guarantee you seeing snow but not having to stay in it. In Flagstaff, if there's enough snow there's even a ski hill and you could ride the chairlift to the top (Brrrrr) and nearby is a snowpark for sledding for the kids. I wouldn't advise driving any further north than Cottonwood in winter.
Also, I-40 can be treacherous in winter. We've even encountered it closed because of a blizzard the end of April. Come in via I-10.
If you're driving from the Gulf Coast, two weeks is going to go by fast!