Problem with moving sms

It really shouldn't happen, because the SMS app is just a front end, they all connect to the same message database and use the same messaging service. You have given Next SMS permission to access your contacts? Though that shouldn't stop the messages displaying, it would just stop it showing names for the sender.

Otherwise all I can think of is to force Next SMS to resynch with the message database. Textra actually has an option for that, but that's unusual, so you may have to clear data for Next SMS to force it to restart (which will unfortunately remove any customisations you've made). And I don't guarantee success, this is just the only guess I have.