You are from Az. Did you get an elk tag? Are you hunting another state for elk this coming fall? If the answer is yes then choose a rifle that you could use for those longer shots. If the answer is no get the 308 and shoot the hell out of it.
I use my 308 to practice all manner of positions. With bi-pod prone with rear bag without rear bag, standing, kneeling, etc. I also go out and shoot jackrabbits.
You shot the 300 Rum but did you shoot it multiple times? See if you can shoot a variety of magnums.
I'd bet if you went to one of those shoots that Ron mentioned other shooters would give you a shot or two. I do it all the time for guys at our range.
As for getting a brake for a 300 win mag if you don't want one don't get one. You can learn to handle the recoil. Shooting from the bench generally will beat you up the most. There are things you can do to mitigate the effect. Sit upright with bag and front rest higher. Put on a recoil pad like the Shooter's friend:
The Shooter's Friend Recoil Pad : Cabela's