My iPhone randomly stopped sending iMessages or Facetimeing if it is not connected to WiFi.

My iPhone randomly stopped sending iMessages or Facetimeing if it is not connected to WiFi (works fine on Wifi). I have gotten a new SIM card and upgraded from a 6 to 6S, and it still doesn't work. Cellular data is on. I've tried logging out and back into with Apple ID. Turned both iMessage and Facetime off, then rebooted phone. Turned on airplane mode and rebooted. I've tried everything. Suggestions? Because I got a new SIM card and phone, I'm thinking it may be an issue with my Apple ID? I don't know what, nor do I know what I would need to change. I have even tried restoring my new 6S and set it up as a new phone but when I go to log in with my Apple ID, it kind of freezes. I left it for 10 minutes and it never logged in. That's kind of what is making me think its an issue with my Apple ID...