Textra SMS automatically inserts a copy of all messages into the Android Stock / System messaging database.

Thus you can backup the Android Stock / System messaging database on your old device, using any number of apps that do this from the Google Play store and transfer to your new device.

We recommend SMS Backup & Restore.

Once all your messages are on your Android Stock / System messaging app on your new device.

Resync from Textra > Settings > About > Resync.