Do you have a video? It's hard to tell what exactly is going on without a video.
What do you do to "move him over" and keep him from shouldering in? If you simply turn his nose away from the barrel with your outside rein, that is the wrong way to do it. (This is where a video would be nice to see how you handle him.). You don't want to allow him to drop his shoulder because he'll either 1) hit the barrel 2) turn the backside of the turn wide 3) blow past it. So yes, his shouldering problem needs to be fixed.
Some horses do better when the rider leaves them alone; other horses need guidance during the run. Again, a video would be helpful.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.