Let me first thank all those who have tried various solutions to this mess Apple has created with Message notifications.
And let me say that I tried disabling iMessage as a fix, as suggested. And then the problem got worse (how could that be possible?). With iMessage disabled on my iPhone, I then got NO messages... period... on my phone. No text notifications, and no messages showing up, with either the phone locked or unlocked. Yes, SMS was still enabled under Message settings. Messages and notifications still came thru on my Mac and iPad, but my phone was silent, even with bluetooth off, which used to solve the annunciation problem. Facebook Messenger notifications did come thru, however.
Bottom line, iMessage needs to be left ON on my iPhone to get texts.
The mystery and frustrations continue...