NAP,
I can't comment on the 338-06, although looking at it on paper gives it a certain appeal.
I can give high endorsement to the 350, as I am the proud owner of one, a 700 classic, that is my favorite rifle bar none.
This rifle is versitle, accurate, not too kickish (reminds me of a heavy loaded naught 6), and the classic handles better to me than anything.
Versitility wise, it will literally handle everything on the North Amercian continent, and more than likely everything on the African continent as well. 250 grain heavy wieghts for Big bears, 225 grain for pigs, elk, small bears, 200/ 180 grain for deer, and 158 grain pistola bullets for varmits. Most of these are good out to 300 yards, and the worst will shoot 200 yards all day long.
Accuracy wise, the best load my rifle has shot was a 200 grain Hornady over a max charge of 3031, grouped at less than .4 inches center to center @ 100 yards. This same bullet will go 3000 fps, according to the Accuratte Arms manual, a fact that I verified with the very fellow that did the testing, who is a co-worker of mine.
I'm none too sure how recoil will be in the model 7. My model 7 is in 308, and it thumps me about as hard as my 350 does in a model 700. Recoil is only a factor when shooting off the bench, so I would advise using something like a PAST recoil pad, as I do it with the smaller calibers as well. Aniticpating recoil is about 95% responsible for not getting a surprise break, and it will effect your accuracy testing like you can not imagine.
I think you will like it, I am real happy that Remington has brought it back, as I was pack-ratting all kinds of brass, and factory loads, most of which had been on the self for years before I bought it.
You have a couple of real good options, but I don't beleive you'll be disappointed with a 350.
Good shooting,
Steve