I drew the rifle hunt there several years ago. Never saw a ram in my compartment, killed an old ewe. Archery hunting some of that stuff would be next to impossible unless the weather conditions were ideal, and even then difficult. I hunted hard for 2 days before even seeing anything.
It would be a very tough hunt for sure. I talked to one of the biologists, and he said that one guy last year killed four aoudad - including a pretty good ram. He found a trail they were using to go to the river to drink and just found a good hiding spot next to it. He kept shooting as they went by. That turned into some serious work, I imagine.
Wow. Every draw hunt I've been on, they always require you to bring everything you shoot back to the check station - hogs included. I grew up hunting them in the Panhandle. We ran cattle on a big ranch on the Caprock. The canyons were full of them. I've had aoudad cooked every way imaginable, and it's horrible table fare. I've eaten young ewe to old ram. It's just different degrees of tough and nasty. Heck, I've seen carcasses rot, untouched by coyotes.
The only reason to kill one is to mount it, IMO. I don't like to let meat go to waste, but aoudad is one animal I won't eat. I hear people occasionally saying how they love it. More power to them. It's nasty. 
