You can get highly varied conditions and weather in the SW about that time of year. It can get to be over 100 degrees or snowing. I'd say AL should be O.K., although do not hold me on this. Obviously there can be rain, then freeze, so ice will form. I'd speak with the Zion bc office.
I was at the South Rim one New Years and it looked fine. I saw backpackers coming out of the canyon, and many will say cooler hiking is much better. Bryce may still have snow, it does depend. At 8,000 feet elevation, it gets cold at night.
You can get high winds, unpleasant then for any outdoors. I'd be open in my plans, and ready for alternate destinations. There's Arches and Canyonlands, and GBNP, Southern CA, and elsewhere.
Have fun and good luck!