There are several things you can do to reduce the recoil on a rifle. One you can get a limb saver recoil reducing pad. They sell them that will install on most any popular rifle. Or you can go with the generic one that slips over the rifle's butt stock.
There is also the P.A.S.T recoil system should pad that you can wear. You slip it on and it has a fastener that goes around your body to hold it in place. This works well even with a limb saver pad. I bought one of these to help absorb the recoil of my 300 Win Mag. I broke my collar bone and while it had healed, the Dr said I should take it easy shooting rifles.
You could always install a muzzle brake (I hate them) and that would reduce the recoil down to less than your 30-06 with some systems. They are extremely loud though and that is the major downfall of them.

As to how does a 12ga 3" turkey loads compare to shooting a 300WM or 338WM? I would say the 12ga certainly is stout but it's recoil impulse isn't as sharp as a rifle. More of a hard push. I don't find the Recoil that bad on the 300WM with 200gr bullets. The 338 has a little more recoil with the heavier bullets but it still isn't that bad.
Currently I am shooting a 350Rem mag in a 6 pound model 7 and it kicks just as hard(perceived) as the 300 or 338 when I am shooting the 250gr bullets.