FWIW,
The last time I flew out of LAX (Los Angeles) with a shotgun a few years ago, there was nothing dicey about it. I looked at the TSA rules online and the airline rules, before I went to the airport. I didn't fly with ammo. The gun was in a cheap Docskil plastic case with very cheesy locks built in to the hasps. I flew directly into Newark. There was no problem in Newark either.