Your 06 with a quality bullet will do just fine. As would your thought about a .35 whelan...fine moose, deer, bear,etc choice where ranges are 250ish & under. A friend just used his 7600/35 whelan to take a moose last week.

I have used a number of cartridges to harvest moose & had great success. I use premium well constructed bullets such as Partitions, barnes, etc. However I made the decision a few years ago for a dedicated moose/elk gun. I chose the .338wm mainly b/c of its adaptability for NA game animals. I have also always liked/wanted one. I plan to hunt Yukon moose and the grizzly bear card solidified the .338wm as my choice. I have killed 2 bull moose, 2 bull elk & 2 deer with my handload 225gr Noslers flung from my .338wm. Awesome terminal performance and no tracking has been required.