I can't speak about the case feeder, but I too have had priming issues with my LnL AP.
You say the slide moves, but it doens't drop the primer onto the primer ram, correct? Does a primer fall into the shuttle, or does it move back and fail to retrieve a primer? If it isn't retrieving a primer, the shuttle isn't moving back far enough. The timing (cam) rod may need to be moved further to the rear to achieve enough movement. The timing (cam) rod may also have become bent, making it impossible to move it far enough to the rear to reliably retrieve a primer.
Does the shuttle move all the way forward with a primer? Sometimes I put just enough forward pressure on the lever when trying to load quickly that the primer ram prevents the shuttle from moving fully forward to present a primer for seating.
I have also had problems with the primer tube backing out of the primer body, allowing primers to flip or turn sideways. When they turn sideways the shuttle becomes stuck and must be cleared.
Make sure your shellplate is clean, tight, and free of any obstructions in the grooves for the case rim. Maybe all your problems are from an obstruction in the groove that isn't letting the case fully seat in the shellplate.