Gulf Shores, AL is too cold for me during the winter. November thru mid-December might be an option or maybe late March to early April, but you get a lot of spring breakers during that time.
Our first year we left SW Michigan on February 1st. That was much too late and have left earlier the past two years, leaving December 30 this year. Next year it will be a tad earlier and stay a bit longer.
If you're going to AZ stay south as previously mentioned. NM is too cold other than passing through to TX. We've been going to FL the past 3 years and love it, but stay further north (Crystal River) because there's less traffic, it's a bit cooler, and the prices are much much better than south of Tampa.