Are you able to activate your iMessage and FaceTime? Can it be that your iMessage and FaceTime are not activated and that you are having activating issues? I ask because the above justify all texts going through SMS.
For your information and just so you know, iMessages always go out in a BLUE bubble while the SMS messages go out in a GREEN bubble. You said that you are charged for SMS messages and this is one way to know whether your messages are using iMessage or SMS.
If you are having trouble activating your iMessage and FaceTime then click on this link for more information If you get an error when trying to activate iMessage or FaceTime - Apple Support
Axel F.