The ST is tourable year round. You'll need clothing/bedding in winter for temps into the teens at higher elevations, and good rain gear. Be prepared to wait out severe wx, mostly in the mountains of NM. Just prepared as it probably won't happen. Pay close attention to wx forecast. High/lows here.

Rather than carry the needed very cold wx gear the entire distance, consider mailing it ahead to somewhere in central Texas and ditching it in AZ.

Cuddle with your water bottles. They will freeze overnight and may not thaw all day. Get yourself warm gloves, wool socks, watch cap, etc. Layer up in the morning and shed as the temps climb. Plastic grocery bags make for cheap, water/wind proof shoe booties. Tap the Warmshowers network, esp The Pecan House in Los Cruces.