Your first and formost responsibility is to your children. Do you want them around your kids? I would not continue this relationship if I were you. Less than three months and so much turmoil. Drunken rants, dramatic ultimatums, childish break ups. Are you honestly asking if you should stick this out?
They are not ready for this. Let them cut their poly teeth on someone else. Bail out. Change your number. This isn't healthy. It's affecting you badly and it will eventually affect your kids if you keep at it.