Well, you can't "show up" in NM for deer, but after the draws are over (mid to late April) the Game and Fish will post a list of leftover licenses (in July-ish) that will go on sale shortly thereafter. You can login online and buy one "over the counter." These are usually in the less desirable units, and often using bow or muzzleloader, but they can still be good hunts. My best mule deer in NM was taken with a muzzleloader on a leftover tag.
We also have "over the counter" tags in some units on private property, but you need to make arrangements with the landowner in advance to access that option.