iPhone iMessage issue

[tl01]

My friend has been using a very damaged 5s for a while as her second phone. For months it hasn't been able to receive texts while not except when on wifi. The other data components work like email and internet. I was hoping it was due to the phone being damaged. Tonight after a ton of work, we got her set up on an iPhone 6. Sadly, the problem is persisting. I am guessing there is something that is corrupted in her back up from her old phone. I was hoping it was hardware and not software. Could there be a setting or another reason? She would prefer not to lose her text conversations so we wanted to check before we set up the phone as new. Thanks!