I agree that you might set some boundaries with the phone. Perhaps you might ask him to help .you with your fears by losing you access to his texts and being with you away from the phone?

Think a out what might help and ask for it. You are in this together. I'm sure he has flaws too. Tell him that you need reassurance. You should not have to white knuckle. If he is secretive on his phone that is a problem. This is a problem for both of you to solve. Instead of him saying you are pushing him away ask him to help you? Just a thought.