Re: Shotgun scopes
I suggest you stick with a 1x red dot, such as an Aimpoint. You really don't need to go with a full size, so the Micro would do the trick. You can get one of the MRD such as one from Trijicon or Doctor too. I know of a few that went with the less expensive Burris Fastfire II and it has held up well enough, but I wouldn't go much cheaper due to the high recoil resistance it needs to have when pushing the heavy loads you will be using. I use one on my Buckmark pistol, and it is a pretty good sight. I would recommend getting the best one you can afford, and make sure it has a decent level of water resistance.
Not sure what you have in the lines of mounting options on your existing shotgun, but you might want to look at replacing the barrel with one that has a connected 1913 rail on it. That way you can use that setup for the turkey and deer, and go back to the other barrel for regular bird hunting without having to worry about zero shift.
Hope this helps.