I'm from NM. You didn't mention the cost factor, but if you want a guaranteed chance to hunt, scan the Albuquerque Craigslist after the draw in NM. Results usually come out in June. We have big bulls here, and if you buy a landowner tag, you're guaranteed to get a tag. A guided trip will just about double your cost. That is out of my price range, but maybe not yours. NM units 13, 15, 16, and 17 are all excelllent for elk.
We went to Colorado near Durango a few years ago on an over the counter cow elk hunt. Lots of hunters, few elk, and small bulls. But, the price was right.