The vast majority of Cooeys were chambered in 2 3/4". If that's the case with yours, a 3" shell WILL FIT into the chamber (because the chamber is 1/4" longer than the unfired shell - to leave room for the crimp to open into when the shell is fired). Although the 3" will fit and Cooeys are [generally] strong enough to keep you from blowing your head off, you shouldn't fire 3" shells through a 2 3/4" chambered gun.
The recoil on a magnum shell SHOULD be greater than a lighter load. But, it shouldn't be as you describe - which is one of the clues that yours is probably 2 3/4". The extra felt recoil comes from the fact that the crimp on the 3" shells has no place to open into except the forcing cone. This unintended constriction of the forcing cone drives the pressures way up (read: bad news). Do yourself and your shoulder a favor, confirm the chambering on that gun and stick to the loads it was built for. BTW. Heavier loads do not equal better patterns or longer effective killing range. That's B.S. Read W.W. Greener's book "The Gun" if you don't believe me.
Oh, and just because I'm a "grumpy old man", as the wetbacks on this site call us, I feel compelled to tell you to drop the "shottie". A shottie is a one ounce shot at a bar, ordered by an 18 yr. old girl or a ###. What you bought is a gun (you don't even have to call it a shotgun because, if it's a firearm and it ain't a shotgun, it's a rifle, a pistol, or a revolver. The term shotgun or simply "gun" refers to a smoothbore long gun.)