Wirehair;
Where are you getting a 5.56x52?
COTW and Donnelly do not have that cartridge listed, 5.56x57 is the closest thing to it.
5.56x52R is the .22 HP, it was developed from the .25-35 case by Newton.
It has a rim and it headspaces off the rim, not the shoulder.
If you are making cases from .30-30, you may have problems with thick necks, you may also be work hardening the case by the major downsizing. You should anneal the cases.

25-35 cases are the best to start from, or use .225 Win, cases, the neck is already 22, the rim is a little smaller but plenty big enough, just run a .225 case in a .22 HP resizing die, load a moderate load, and fire form the case.
Don