Agree. I'm fairly sure they have a shuttle as well, but the walk is quite manageable.

It's been nearly ten years since I've stayed at the Hilton Garden Inn on Yale, but it wasn't bad. It's just a little bit farther from the terminal than the Sheraton.

OP, I'm sure you're already aware that it's over a four hour drive from Santa Fe to Carlsbad. IME, there aren't a whole lot of places to stay near Carlsbad Caverns. You might have better luck in Roswell (which has its own quirky charm). 👽