Odd going's on there with the case not fully entering the die.
I would suggest that you:
Pull all of the 'guts' from the die leaving only the die body. Then attempt to seat a case fully into the die. If you cannot, then there is something wrong with the die. If you can, the most likely problem is setup.
Start by screwing the decapping rod back into the die body to a depth of only two full threads. Run an empty case back into the die. If there is no problem, leave the case in the die, and screw the decapping rod into the die until you get resistance. Then back the case out of the die, screw the decapping rod in until the decapping pin extends no more than 3mm from the bottom of the die body. Try to seat an empty case again. If you have problems, it's most likely the die. Take it to an experienced reloader and have them have a look at it.
With respect to neck-only sizing, some 30 years ago I bought into the idea that it would produce longer case life and better precision. Those suppositions make sense. However, after many years of using that method for many cartridges, I have completely abandoned it because it did not produce the desired results for me. I don't have any problems with case life WHEN FULL LENGTH DIES ARE PROPERLY SET UP FOR A SPECIFIC RIFLE'S CHAMBER. Furthermore, I realized NO improvement in precision.
I'M NOT SAYING OTHERS MAY NOT HAVE REALIZED THESE BENEFITS. I AM SAYING THAT AFTER EXTENSIVE USE AND A DESIRE FOR THIS PROCESS TO "WORK", AND IN SPITE OF THE "LOGIC" OF IT, IT DID NOT PRODUCE LONGER CASE LIFE OR BETTER PRECISION FOR ME.
If you (the editorial you), wanna neck-only resize, and you get both longer case life and better precision, I am nothing less than 'pleased as punch' for you. As for me, I have quit parting with the extras expense of neck-only dies and the hassle of having to segregate all of my .308 Win reloads (and other multiple-rifle ammo I use), depending on what rifle they are to be used in.
Regards,
Paul