Re: lee full length sizer in .223 question
Not sure I understand what you mean but Lee's FL dies are like all other FL dis so far as set-up goes. No FL die goes "to the base", they all stop 1/8" (.125") off the base so there's room for the shell holder.
No die "instructions" can do more than get us in the ball park; correctly adjusing an FL requires tweaking (ditto seaters). Die and shell holders are tightly machined but they all have slight manufactoring tolerances. Chambers do too so getting the best cartridge-to-chamber fit has to be done carefully by trial and error, not by following a formula.
That said, the best common starting place will have the bottom of the sizing die screwed down to contact the fully raised shell holder plus an additional 1/4 turn to compensate for press deflection under load. If you really have a 1/4" gap between your die and shell holder with the ram full up you clearly don't have the die screwed down far enough.
If you actually do have the die screwed in correctly and it gets really difficult to push the case in further then you don't have enough case lube applied, especially on the lower, thicker part where it's hard to resize. Use any commercial case sizing lube and follow the maker's instructions, all should go well.