I have also found through trial and error that the Lee dies just don't work as well on the LnL-AP as Hornady or RCBS (or any other dies with much longer threads and die bodies).
You can make the Lee dies work, but they aren't as solid, don't have as much adjustment and don't produce the repeatability that other brands will.
I've pretty much switched over to the Hornady ND dies over the last year. Better quality than Lee, work better with my press and having the ability to add micro adjustment to them is nails. I still use the Lee FCD dies on my .223 and .30-06 as their collet style FCD's are better than anything else on rifles.