Is this normal of a gun to move in its stock?

It's normal, but not what you'd like. You can try bedding it better. But you may still have problems. It's quite possible that the decreased accuracy is from the fact that the barrel is free-floated. There's a myth that free-floating always improves accuracy. Remington disagrees, and puts a pressure point at the fore-end to dampen barrel whip. Bed first, and make sure everything else is up to snuff, but you may wind up having to replace that pressure point. From personal experience, I can tell you that it's a tedious job.