You do not shim a barrel on a Mosin, you shim the action tight to the stock.
The Finns used square brass shims with the hole in the middle for the action screws(reciver /mag tang screws) and tighten them.
The Mosin Stock is long and thin on an M-91/30, something the finns didnt like either. If its slapping the barrel during recoil, it could be a problem. Do you have the cleaning rod in place?
Is it a consistant 6 inches, a nice round group or is it a strig , up and down on the target.......could be stock slap or inconsistant ammo.
Raising your front sight will lower your point of impact. Some trade off front sights, some put a small tube thats a bit longer, some put in their own pins, made a bit longer. Also your sights are most likely set for 150 grain "light ball" ammo, if your shooting milsurp through your milsurp.
Most 91/30's hit high at 100 meter or yards, they are generally battle sighted for point of impact around 150 to 200meters, depending on where it was built, and hitting high wasnt consitterd a problem. The soviets aimed for the belt buckle.