I have been hunting them every year in Arizona for the last 18 or so years and find it to be one of the funnest hunts there is. We usually drive the washes and look for tracks and then head to a high spot to glass for them. Or if we are not seeing any tracks we'll just get high and glass and then do the old spot and stalk. So it all depends on the type of area that you are hunting in. I know that some hunting in Texas is done around feeders that are set up for other animals but for me that isn't hunting.
As far as field care they are no different than any other animal except for their sent gland that is located on their back. You don't want to mess with it and it comes off when you skin them. Now for cooking, I like to smoke them using mesquite wood. I'll usually smoke the hinds for about 9 or 10 hours at a low heat and then let it sit overnight. Then slice it nice and thin and feed it to people that have no idea of what they are eating. I usually disappears fairly quick. I have a friend that will fry it up with peppers, and onions and enjoys it that way. There are others that will make chili out of the meat and others that will BBQ it. In all of the ones that the group that I hunt with have shot there has only been one that didn't taste very good, actually the one that shot it said that it was the rottenest meat that he had tried. But that was from having problems finding it after he shot it and it was quite warm that day.
I've only hunted them once in Arizona like Critter talked about and it is definately the way to do it. If you have the open land in Texas or big ranches the same would be true there as well. Alot of the ranches down there just use the corn feeders and there is no hunting involved to it just wait till they come. Not recommended at all and no hunting fun there. New Mexico should be pretty close to the same as Arizona as far as the hunting goes. I hunted mine right on the border there and didn't do too bad.
First make sure your gun is safe, remove the bolt and clean your barrel from the chamber to the end of muzzle. Clean the barrel using butchers bore shine until a patch comes out white. Run a total of 5 separate patches soaked with denatured alcohol through the bore and let dry for 30 minutes. Insert a cleaning rod with a jag and a dry patch into the neck of the chamber. Make sure it will stay there through the next process. Now using another cleaning rod, install a patch that has been coated...