Recoil is primarily a factor of the weight of the shot charge compared to the weight of the gun (with a nod to the speed to which the shot charge is accelerated). All things being equal, if you reduce the weight of the gun (short barrels, for example), you increase recoil. Again, all things being equal, if you increase the weight of the shot charge, you increase the recoil.
So, if you're concerned about the recoil, add weight to the gun. Adding weight near the muzzle makes the gun harder to accelerate and decelerate, thus harder to handle. Adding weight between the hands or to the butt stock has much less deleterious effect on handling. Many side by sides have a through-bolt securing the stock. With these, weight can easily be added by simply removing the butt plate and inserting lead or other weights into the bolt channel. You can also bore holes in the butt stock under the butt plate on guns not designed with a through bolt. Weight can easily be added temporarily as well. One of those side-saddle shell carriers over the stock can work and be removed easily when desired.
The other side of the equation must be considered also - the weight of the shot charge. All things being equal, a 1 1/4 ounce load will generate more recoil than a 1.0 ounce load.
Rather than worry about 12 ga vs 20 ga, pick the gun that has the characteristics you want (single or double trigger, barrel length, available shells, etc) and select shells with the appropriate shot load, then decide if and how much weight needs to be added to the gun.
I'm not up to speed on cowboy action shooting rules, so there might be limitations to your options that aren't covered here. If you don't plan to do CAS, this info should help you get squared away.