Jeff,
Mesquite is one of the most stable wood species. It shrinks very little from harvest to kiln dried. (See
http://www.planethardwood.com/documents/WoodStabilityChart.pdf) Many here in Texas would turn it very green to finished dimensions without concern about distortion when it's dry later.
Any finish is good on mesquite, but my preference is Waterlox "Original".
Laurence,
The "hammerhead grubs" you spoke off are probably mesquite borers which are a long-horned beetle in the family of Cerambycidae, not a moth, which is in a different Order, Lepidoptera all together (beetles being in the Order, Coleoptera). (See
http://www.texasento.net/Placosternus.htm) Trust me, I was an entomologist before I was a turner.
I was once cutting up a mesquite log in the Texas Hill Country, and out of nowhere came dozens of these beetles and they started laying eggs, which are not visible to naked eyes, esp. on the (very) rough bark of mesquite. It is the larvae that hatch from these eggs that bore into the wood (mostly just the cambium layer and the sapwood) and do the damage. Once they bore into the wood they are essentially protected from many treatments. One south Texas mesquite dealer told me that he fumigates his logs, which is effective. It is a commercial operation that requires a license and most of us turners can't do.
Most plants have natural protective mechanisms against natural enemies when they are healthy. When they are weakened by stress of one sort or another they become susceptible to attack by these natural enemies. In the case of mesquite not only does the damaged/weakened wood become susceptible to the beetle attack but there might even be a kairomone that attracts the beetles, judging from the short time from us cutting the log to the arrival of the beetles.
The bottom line, Jeff, we have no idea if your mesquite has any beetle eggs/larvae inside. The safest thing to do is to turn the blanks right away. That way even if there are larvae inside, the damage is minimal and might not be visible. Since mesquite is stable, you have little concern about distortion when it becomes dry. Oh, because the wood is stable, checking is not likely, unless you have "ring checks" to start with, which is common in mesquite for some reason.
Mesquite ranks high on many turners' list, me included, and some has a pleasant aroma. So, enjoy turning it.