We have been all over UT in January, you have to get real lucky in January. High elevations like Bryce Canyon are out.
We stayed at Snow Canyon SP in St George last February. January and February are the perfect time to visit Zion IMHO. Even there, you may need to winterize, it really depends on the weather. They had snow in St George the week before we arrived. Its typically dry in St George, so the snow is not going to stick around.
Average low temperature in January for Moab is 20F. Average low in St George is 31F. Big difference!
I70 west of Moab goes as high as 7,900 feet elevation. Moab is at 4,000 feet, and St George is at 2,900 feet. So you are likely going to see cold weather on I70. If 4,000 feet is a low of 20F, then 7,900 feet is going to be more like 10F or lower.
My advice, unless you get super lucky temperature wise, I would not plan on Moab.
You are lucky to be coming from California. Coming from NC, I had to really be careful on routing. Ended up taking I10 all the way from Mobile to Phoenix. Route home in early March we hit another cold spell. Amarillo, TX was 15F, I was hoping to take I40 home but it didn't work out.