I've had 338-06 in the past. Keep it 30-06 and load 200-220 gr bullets for it. You'll get much better performance than 338-06 with 200-225's and save a lot of money. Go heavier than 250's in the 338-06 then you MIGHT see marginaly better performance, maybe. Unless it is already 18.5" cut it no shorter than 20".
And I'd do it in a bolt gun too. I like a pump action shotgun, but rifles aren't used like shotguns.