A coupla options...
First,make sure your stock fits.
Second,learn to shoot from the other shoulder.
Third,get the forcing cone done on your present 12 ga, drop down to the reduced recoil loads,heavy up your gun, or do any of the various recoil reducing things old threads are filled with.A bit of research in the Archives will help you.
Fourth, get a gas gun,and do the above.
Fifth, try a wearable recoil pad, like the PAST, for practice and competition.
I wouldn't recommend a 20 ga for less kick, some of the lighter ones are vicious kickers. And if the shotgun has to be heavy, might as well get a 12.
HTH...