You can check ammo in the original boxes, up to 11 pounds, IIRC. It's a good idea to go to the airline's website and print our THEIR rules and policies.
From what I can see, AZ and NM are good to go, but check it for yourself to be sure.
http://www.handgunlaw.us/LicMaps/ccwmap.php or similar.
I have not driven that route in years. Some of it gets close to the border with You-Know-Where. Parts of southern AZ are posted with warning signs but your route ought not take you into those areas, mostly south of Tucson.
Obey all laws. Avoid car trouble. Carry communications if possible, like amateur radio, if you are licensed.
Prior to flying make absolutely certain that you have no firearm, knife etc paraphernalia on your person or in your carry on.
Luckily, I have enough willpower to control the driving ambition that rages within me.