The primer acts just like the rings on a piston of a internal combustion engine and seals combustion pressure inside the chamber. Your getting blow-by caused by a primer that isn't sealing the primer pocket. If the cases had less internal volume your primers would be flatter from higher pressure "BUT" it doesn't mean the higher pressure would cause the primer to leak around the circumference of the primer.
Photos of the primers would help but it sounds like the primers are too small in diameter or the primer pocket diameter is too large.
Below is what happens when people reload .223 cases with loose primer pockets in their AR15 and they loose the gas seal and have high pressure gas leakage.
You need the primer to provide a good high pressure seal, please notice below how the primer can move when the cartridge is fired. The primer can move to the rear under pressure and is pushed back into contact with the bolt face. Then the chamber pressure builds higher and the cartridge case stretches to meet the bolt face.
You need to find out if the primers are causing the problem or if the primer pockets are defective on your cases.