I have to recommend you stay with the 20 gauge, if you are looking for a dual purpose gun. A 20 ga. RB is 3/4 oz in weight, plenty heavy enough for deer out to 100 yds, and even beyond. In shot loads, it is more than adequate for turkey, and other game birds out to 30 yds. If you have the barrel jug choked, you can extend the range depending on how much choke you put in it. BP hunting is not long range shooting- ITS HUNTING! You are suppose to get in close. That is part of the experience, particularly with a flintlock, smoothbore gun. The 20 ga. will do all you need to do without rattling your brains. Unlike using cartridge guns, the secret to tight patterns is not more powder, or more shot. It is less powder for the same amount of shot. When you talk about round ball shooting, and get to the 16 and 12 gauges, and larger, recoil does become a factor, even shooting Black Powder. I would not argue that the guns produce better patterns shooting more shot, but when it comes to shooting round ball, it takes a lot of powder to get those large balls going the same velocity as the lighter 3/4 oz. slug. Even going slower, the 1 oz. and 1 1/8 oz RBs produce noticably more recoil.
RB don't kill by producing massive SECONDARY cavities in tissue like high speed cartridge bullets do, with their trailing vacuums and air turbulance. They kill by making a large PRIMARY wound channel in tissue and organs, flatting out on impact to expand and make the wound channel even larger. All this causes death by hemorrhage, much like a modern arrow does, if the ball does not hit nerves and kill instantly by shock. A .62 caliber hole through a deer that expands into a larger primary wound channel causes a lot of tissue destruction, and makes it much more likely that some nerves will be struck stunning or killing the animal outright.
You can load non-toxic shot to shoot Migratory waterfowl in these guns, but you will work to get the birds inside 30 yds for a clean kill. That is probably better, as too many hunters shooting breechloading shotguns don't have a clue about the effective range of their guns, and are trying to kill birds at 80 yds. and beyond. At least when you know the limitation of your gun, you discipline your choice of shots, and actually will kill more game.