I've used a .338 WM on several deer, all mulies. My pill of choice is the 210gr Nosler Partition. Meat damage is minimal. Most all are one-shot kills, with only one exception and that was my fault (didn't dope the wind correctly and hit the deer too far back).
In all cases where I put the shot through at least one front shoulder, the deer never took another step.
Now, as to recoil: Expect significantly more recoil than a .30-06. And if you use a muzzle brake, always, always, always use hearing protection. If you do not use ear pro, you will lose hearing from even just one shot if you have a brake on it.