If you spend much time talking with hunters going for their sheep grand slam, you will quickly learn that the Desert Bighorn is the toughest to draw and/or most expensive tag to purchase. If any sheep species will do to get your feet wet, then I would focus on the Rocky Mountain sheep. More tags available.
It's true that sheep will be very difficult to draw in AZ. The odds are definitely against you. You would have little to lose by applying, except for the fact that you have to purchase a NR license to even apply for the tag each and every year. It is a costly proposition to be sure. But bear in mind, residents have to purchase a license as well just to get into the draw.
As for Az draw being all jacked up...... if anybody can come up with a better way, please step forward. Az does not have the amount of favorable habitat that most other states have - be it for sheep, elk, deer or antelope. We have a lot of desert with no water. We have a thin strip of forest that runs southeast through the northern third of the state. Approx. 30% of the state is tribal land and not available for hunting except for those with a fat wallet.
Our elk herd is estimated between 20k-30k, while CO has over 300k of elk. And yet, AZ is a destination state for trophy desert bighorn, elk and mule deer. With more than 3 Million residents in the greater Phx area alone putting pressure on these limited resources, this ain't no accident. AZ Game & Fish have done nothing short of a miracle to keep hunting opportunities open for as many as possible. Do I like the institution of a draw? No. Was it necessary given the limited resources? You bet! Look how many other states have had to follow suit.
AZ has exported bighorn sheep to UT, CO and NM. These are sheep that could have been used to increase tags for hunters. But instead, AZGFD has taken a long term approach by attempting to establish herds in other states that will one day be huntable to local residents w/o them having to buy out of state licenses and tags. I hate the wait....but my hat is off to the AZGFD.