How DO you make your civilian woodcutters hold onto their axes when drafted? In my experience there is no correlation between whether I assign them to have axes or not, and whether they drop their axes on the ground or not.
Short answer: You don't. At least, I haven't figured out how to. But it's fun to watch a naked woodcutter try to wrestle to death a werewolf that has forsaken its claws in favor of a pigtail glove taken from its first victim. (They went on and on in a total stalemate for several minutes, until the werewolf eventually used up its migrating-through-the-map time and ambled offscreen.)
It's part of the Fun.
I've found that Axes get claimed for either woodcutting or military, but not both. So if you want to have your woodcutter join the military, you must first disable woodcutting, causing him to drop his axe. Then, draft him, and he'll pick up his axe and go slaughter things. Once that's done, you have to remove axes from his squad so that he'll drop the axe again, then return him to civilian duty and enable woodcutting.
Yeah, its a pain in the butt. I find it helps to either A) avoid this rigmarole by cutting down a bunch of nearby trees at the start and then disable woodcutting/assign squad axes in order to have your military dwarf do his other duties while carrying an axe he won't drop, or to B) make a weapon stockpile near your dwarf when making him drop the axe so he doesn't gallivant off-screen and aggro while his axe is on a random mountainside.