The M1A barreled action absolutely should NOT have any "springiness" when the action is latched into the stock. Follow Gunnut69's suggestion and look for interference between sling swivel stud/stud and the gas tube. Any movement here and the rifle will badly string shots vertically.
As I recall, the match conditioned M1's I fired in the Navy had the front stock ferrule and the forward handguard epoxied into one assembly, and extra care was given to make sure that the action was secure fore and aft when latched into the stock. The action bedding points were set in fibreglas resin. I'm sure the M1A has a similar arrangement.