Difference between 223 and 5.56 chamber is kind of two areas.
The first one is the throat cut at the end of the chamber.
The second is the side wall dimension of the chamber itself.
So the 223 has a tighter throat (less jump to lands), as well as tighter side wall dimensions (read less blow by at firing forming), and when 5.56 ammo is fired in a 223 chamber these are the the reasons that the 62K ammo (223 is 55K) wants to go over pressure real quick in a 223 chamber.
The solution so long at the chamber has not been nitro treated or chrome lined, is to just take a 5.56 reamer to the chamber to bring it up to 5.56 specs.
To add, depending on the barrel gas port size, could explain the blow primers (over just the chamber over pressure problem) if the gas port is too large and causing over function on it own.