The ACA routing from Los Cruces to Phoenix takes you through some mountinous areas that could be very cold. It'll be doable with the right cold weather gear. You could divert through Tuscon on the interstate to avoid. I do not think bicycles are allowed on the interstate from Tucson to Phoenix so you'd have to plot a secondary route for that section.

The ACA maps will be filled with useful camping information, some of it free, but mostly pay sites. They will not help you with wild/stealth camping, but not likely to take you through any truly 'sketchy' areas. You're not likely to meet many other touring cyclist on the ST in the middle of winter.

You can check historical lows/highs and other info by simply Googling "average weather for ..........." Likely be many nights below freezing through west Texas and New Mexico, but day temps will rise quickly to comfortable.

I do touring in January every year near and north of the ST in Texas and Louisiana and haven't encountered anything not easily managable. Mostly nice cyling conditions. Just be prepared for rain and cold.