What about a savage model 24 combination rifle in 22 mag and 20 gauge? I have a scoped combination rifle 7x57/12 gauge I like alot. I can sit for Deer or Hogs and when a Hare or Rabbit happens by I get dinner even if I don't get my Deer. Mine is really accurate, I can shoot sub MOA groups with it all day long.

Been my experience most times if you miss the first shot the second shot has less chance to connect than the first. My brother was really good with his double shotgun and could reload really quick.

Th Savage with a scope would cover most of your bases, unlikely you are going to find one firearm to cover them all.

You could have the stock fitted and a recoil pad added.

Check local laws and see if combination rifle/shotgun is legal, some places it is, some places it isn't-