Yes. I recommend a hand primer, and moving primed brass to your press. I have used a couple of different RCBS hand primers and they work very well. Hand Primers have a lot more tactile feedback - you can tell by feel when seating properly. You squeeze fairly strongly to seat primers. I made shallow imprints on some rifle primers, and decided that was normal.
Primers need to be seated with some force, as the primer legs will contact the bottom of the primer, and you need to squish them a bit. Primers should be below flush with the case.
If they fire on a second strike they are good primers and we're not seated deeply enough.