This is theorizing, obviously the problems might go a lot deeper than what I am saying, and while much ink has been spilled I don't know if anybody has really thought about this:
A quick idea on what artificially limiting play is and or can be: My roommate played basketball for Goucher (it's a small D3 college) and he was not allowed to dribble. He was the best shooter on the team and so he could only shoot or pass the ball when he got it. So he did and he was a much worse player for it because part of his ability to shoot 3 pointers so well was based off of his ability to drive. There was no reason for him to not dribble but he still didn't anyway because that was what the coach wanted.
Another example, this time with Liverpool (and also an example where it can be positive): Dirk Kuyt is a striker. He scored buckets of goals in The Netherlands and still scores them for the Dutch national team. Yet on Liverpool he plays winger and doesn't score as many goals or play as forward as he would probably ideally like (not that I could ever see Kuyt complaining). Yet he is still the winger because the coach said so.
I think Liverpool are not allowing Andy Carroll to turn on the ball to move forward. With Newcastle, based upon approximately 12 minutes of youtube clips, Carroll faced goal a lot more and was allowed to turn on the ball. What I mean when I say turn on the ball is a holding striker who has the ball and his back to the defender turns around with the ball and beats the defender. With Liverpool when he receives the ball as a forward he is basically asked to then immediately pass it back to the nearest midfielder then get forward for a header. Go look at how he played against Newcastle, he would get the ball 20 yards from goal and then pass it back about 10 yards negating the initial pass. This strategy was not unique to the Newcastle game, but has been a part of how Carroll has played in general at Liverpool. As a Geordie, Carroll had a lot more freedom to turn on the ball to get forward and make plays. Since he got to Anfield Carroll has not been allowed to make plays, only finish them and I think those rules are making him unconfident which is hurting his play.
Go to 57 seconds into this clip and see the third goal against West Ham. (he scored a hat trick)
Carroll makes a great cross-field pass which leads to his goal 10 seconds later. Liverpudlian Carroll would never be allowed to make that pass he would instead be forced to lay it back to either a close midfield player or a defender. At least, seemingly he would based upon how he has played since he moved to the Reds.
Andy Carroll is being artificially limited by Kenny Daglish into being a holding striker who can only get headers from a more complete player who can create chances as well as finish them. If he had more freedom I think Carroll would be play better and score more goals. Free Andy Carroll!