Forget the 338 WM it is neither fish nor fowl and really doesn't do anything very well, it is not as versatile as any of the big 30 Mags (my preference is the 300 Wby, but own and have shot game with the other 2) and it is not a 375 by any stretch of the imagination. If you really want something in between there are 2 that actually do something impressive the 340 Wby (my choice) and the 358 NM or STA, these cartridges make the 338 look pretty sick, which it is in my opinion. The 340 Wby is one of the most impressive killing cartridges I have ever used on game.
Just remember that there is nothing a 338 will do that the 300 Wby won't do better and further away. The 300 RUM is an impressive cartridge but the recoil is a whole serious step up from the Wby, and if you are going to be subjected to that level of recoil, you might as well shoot a 340...........
I believe you said you have an '06, if you load 200 gn Nosler parts or ABs in your '06 with a good stiff load, you would never notice the difference between game shot with that and game shot with a 338..........seriously !! Anyone who says differently is wishing, I've had 3 different 338s and not one impressed me as a good killing cartridge, and not one killed as good as my 300s have, and neither do they have the range and trajectory of the 300s. I just can't yawn wide enough to express my total experience with the 338.
I would take the 350 RM any day over a 338 and have in point of fact, two 350 RMs in my collection and zero 338 WMs............