Hey Coach!
Mesquite is my speciality, although I turn pens out of other woods, since I'm a Texan too, mesquite is my favorite.
Not to discount anything that has been previously said, you have been given great advice, however, mesquite can be turned with moisture in it with no ill effects AFAIK. My very first pens were turned out of "seasoned" mesquite logs from a local firewood dealer. The firewood guy said that the wood was cut the previous year and stored outside in Pearland (close to Houston) in other words as humid as a rainforest! LOL!
Trust me, I didn't know what I was doing then (still don't [
] ) but I didn't have any problems with cracking due to drying. The pens that I made 7 years ago haven't cracked yet and they were turned green, in fact those pens turned my hands purple before I applied the finish. I still get my mesquite from the same guy.
One thing I have noticed with mesquite is the drier it gets the more prone to blowing out it becomes. Turn it fast, use SHARP tools and take shallow cuts. When you turn it with a little moisture in it it seems to hold together a little better.
Lastly, mesquite is just plain beautiful!
Please, anybody correct me if I'm wrong I am just going on my personal experience. I will say this though that the resident turner at the Rockler store I go to confirmed that mesquite doesn't need to be "dry" before working it.