Have you looked at renting a house or condo from a private party? We've used
www.VRBO.com at least a dozen times and are always happy with our rentals. You have to pay attention to what you want / need and look at photos for dozens of listings before you settle on something, but all the effort has been worth it for us in the past.
IDK how old your kids are, but I've spent 3 New Years' Eves at the bottom of the Grand Canyon, and it's warm enough to camp down there! You have to apply for a permit months ahead of time, but I remember applying in October or November and getting approved one year. You could stay at Phantom Ranch, but those reservations are harder to come by. The hiking at the bottom of the canyon in December is great because the weather is so nice. You might get some snow on the rim, but it gets warmer as you drop in elevation.
Canyonlands is another option for winter camping -- the Island in the sky (?) area looks similar to Grand Canyon conditions.
We also spent some time at Zion after one of our New Years' Canyon trips, and it was great. You can't go way in to the slot canyons, but there's still a lot of hikes open. It's cool to see the waterfalls frozen over. We stayed just outside the park, in Hurricane, very inexepensively (but I can't remember how much we paid). We were poor grad students at the time, so these were all low budget trips :)