I would think a 30-06 would handle all of those game animals, but I have not hunted them. Have you looked into threading your rifles for a muzzle break or adding a limbsaver or pacmeyer recoil reducing butt pad? The 300 win mag is going to hit your shoulder harder than both the 30-06 and 308, simply because it launches a heavier bullet faster.

A muzzle break will take a crap ton of recoil off of your shoulder. It does make the rifle louder so make sure you have ears in. Several companies make electronic ear plugs or muffs so you can still have situational awareness while hunting and protect your hearing. Orrrrrrr..... suppress it if your state allows for suppressed hunting.