Well, first of all, what you need more than anything (more than any particular caliber) is a very accurate rifle, to hit your target at distance. I know that sounds obvious, but maybe it's not. So the RIFLE selection and AMMO selection is far more important than the caliber selection. Which RIFLE(s) are you considering? This will tell me what the caliber options are, by the factoring chamberings offered. Are you buying used or new only? Are you unduly limiting yourself to the selection at one particular store? If so, there is no need to when you just buy whatever you want on the net and have it shipped to a local FFL dealer.
As for caliber, I'd choose neither of those two for Southern N.M at any reasonable hunting range, but I supposed if made to choose, I'd say .308 win for distances out to 300 yards, and I'd say .300 win mag if you want to push the envelope and shoot to 400 yards or so (due to better wind-bucking ability of the .300 mag). Or split the difference in .30 cal and get a .30-06. If I had to choose for you, though, I'd go with .308 because there is a lot of good high quality ammo available - if it's good enough for 1,000 yard shooters, I'd say it's good enough for any long range hunting you'll come across.
But also, without knowing more, if you're willing to flex on caliber choice, the best CALIBER for your criteria is probably the .270 Winchester, .280 Remington, .30-06 Springfield, or 6.5x55 swedish for those criteria & game. .300 maggies are overkill even for elk IMO. But again, getting an accurate rifle (and your skill) is more important than caliber here. Another good and very popular long range caliber to look at is of course, the 7mm Rem. Magnum (aka .280 rem with extra recoil).
To me, the theoretical ideal for your use to include Elk is the
.280 Remington, but factory ammo costs a bit more than .270 win or .30-06. And there's not more than a scrunthair's difference between the 3. Bottom line is, they will all work. But the .300 mags work
and also give you a lot of extra recoil, noise, and ammo expense.
Also, it doesn't sound like you're wanting to get into crazy odd overbore wildcats and such, but if you were, the 7mm "super mags" like 7mm STW and 7mm Dakota are popular with extreme range hunters out West, in my understanding, due to the optimized BCs of the 7mm bullets. The .264 Win Mag is another overbore but good longrange hunting rifle.
Your action choice is superb/ideal. What's your budget for this rifle? I'm interested to know your rifle choice.