I tried the punch down fold upside down (so the tight folded part was pointed down towards my tailbone), and that made a world of difference! So much easier to insert too, and I was able to move my finger around the cup to make sure it was popped open completely. Sometimes, I have to really work with it to get it in far enough, but when I'm successful with it, I never have any leaks!
So thanks everyone for the advice and support!