How butt heavy is it? If just a few ounces you could have a bit of wood removed from the butt to rebalance the gun. Quick way to check it is use a rare earth magnet and weight on the barrels to balance it at a heavier weight. Then see if it shoots better. Ive seen guns with weight added to the for end or even to the barrel to get the balance right as a permanent fix. Drilling out the butt stock has limits for the weight you can reduce. Much past a few ounces might be the limit.