Josh,
I use the LNL AP also for 9mm, and none of the stages should allow your case to lean. I've called Hornady and talked to their tech guys a few times since going from their single stage to their AP press, and they are VERY friendly and helpful in diagnosing problems. Give them a ring and I will bet they can help you figure it out.
PS: it took me a long time to find the same comfort level with my AP as I had with my Classic press. You will get there.
Regards,
Tony