Just bought a 1000 Fed Match primers because they seem to have a good rep. I'm going to prime some using the press instead and see if that eliminates the problem. I've had problems with the Lee hand primer before. I hate squeezing the hell out of it, makes me think I'm crushing the primer or have one in sideways or other mishap. And with the Lee hand primer (probably true of most priming mechanisms), once you partly seat a primer, you can't pull the shell out and examine what up, such as a primer flipped or on its side.
Edit: For clarification, this is new brass.
I just seated a couple of the same primers using the press instead, and it does take a lot of force to seat them flush. No wonder the hand primer didn't get them flush. I then fixed the 5 that I'd done with the hand press by seating them with the press itself. I'm done with the Lee hand primer until further notice. High primers isn't acceptable on a rifle where there is less than .001 excess headspace. I'll have trouble closing the bolt and discerning the cause with any high primers.