You don't need much lube, but trying to use as little as possible can lead to NOT using enough. I use a light spray coating and don't worry about removing it--Never have in 40 years of reloading and I figure it almost all comes off with all the handling the case/round undergoes.
Lube only needs to be on the body of the case, not the shoulder or neck.
I assume the stuck case came out in one piece and didn't leave any parts in the die? I assume the removal of the stuck case did not damage the die?
You haven't reset the depriming pin so high that the expander plug is interfering with the case sizing section of the die?
There just aren't that many things happening in a sizing die and an inspection of the inside should show any problem.
I have never needed to lube a case neck, but many people seem to. If I was lubing the case neck, I would use graphite or mica--certainly not any oil/grease.