Is the gun provided with choke, or screw-in chokes? If not, and its cylinder bore, you can file the muzzle to change the POI for each barrel. to bring them to POA. If the left barrel is shooting low, then file the bottom edge to bring it up to POA. If its shooting to the Left of POA then file both the side and bottom to bring it to POA. File at 7:30 on the barrel looking from the rear of the gun. If you are looking down at the muzzles, then file at 4:30 to get the same effect. This is trial and error filing, that is done at the range where you can pattern test the gun as you file.
Do one barrel at a time. There is a learning curve, if you are not used to draw filing metal. Take your time, and don't take off too much metal before checking to see what is happening with the POI as a result of your work.
The right barrel that is shooting right, and high, needs to have metal taken off the barrel at 2 o'clock to bring it down and back to POA. If you are looking at the muzzle as you would be filing it, you are working on the LEFT barrel at 10 O'clock. Got it?
TIP: If you set paper out at 10 feet, you will shoot ragged holes, but they will hit off the point of aim just as your longer range patterns do. The holes will just be smaller, and not use up so much paper per shot! Get the two barrels so that they are taking out the center of your aiming point. Then move the target back to 30 feet, and check to see if the patterns are still hitting POA, or you need to file some more. At 10 yard, you will have about a 12 inch pattern, much smaller than the larger patterns at 25, 35, and 40 yards. I find its easier to see the center of the pattern at these close ranges while I am filing, so that I can get where I need to go faster in adjusting the POI. Once you get close, then you take the target back to your normal patterning distance, and test the barrels at those ranges. ALWAYS DO THIS OFF A REST, so you eliminate human error in tuning the barrels. It can be a sitting, or standing rest. Just eliminate your own errors, and let the gun speak for itself.
TIP #2: Before doing any of this, I would adjust my powder charges up or down to see if I can't get the pattern closer to POA that way. Its not uncommon for individual barrels to require separate powder/shot combinations to get good patterns out of a DB shotgun. The first barrel you habitually fire is the one you want to be closest to the POA. That is going to have the tightest pattern when you slap the trigger. The second barrel will only be fired when you miss your first target, or if two birds rise at the same time. Your pattern will be more open, and the distance to target greater for the second shot. That barrel can be a little off, and still kill birds if the load, and choice of shot is right.
Forget about speed reloading shotguns. It can be done, but always produces an inferior load to shoot. Think of BP shotgun shooting, as taking your reloading tools into the field with you. Do it right, and you will get great results. There should be no rush to reloading a shotgun. Take that time to enjoy the smells, the scenery, to relax your eyes, and ears from straining to see and hear birds, and just enjoying that you had an opportunity to burn powder at a game bird. If you brought down the bird, congratulations to you. All birds taken with black powder guns are trophies.