I started having a similar problem with my LNL-AP after loading several thousand rounds. First it stopped feeding the last primer in the tube. A little later it would fail to pick up primers for no apparent reason.
The first time I called Hornady they suggested I loosen the top bracket on the guide rod and pull it away from the press-that way the primer shuttle would come back a little farther. I did this but it did not solve the problem.
As I was puttering around with it after a stoppage I noticed that the feed tube had raised itself up. I called Hornady again; the tech suggested that the guide rod may have gotten bent and that I should bend it back. I tried bending it a little and it worked better but near the end of my last loading session I noticed that the primer seating was not as smooth. It may be time for a good cleaning of the priming system.
I am going to try shimming the primer feed tube with a bit of tape and see if that helps.