I always use the follow method to transfer text messages from iPhone to computer
Step 1. Choose iTunes backup file for your iPhone
Make sure you have made the backup of your iPhone to iTunes already. Launch the installed software and it will automatically scan your computer and show all the backup lists you have ever synced with this computer before. Just select your iPhone backup file and click "Start Scan".
Step 2. Scan for the iPhone text messages you want
When the scan finishes, the software will display all iPhone files. Find the "Messages" category and preview all the text messages in your iPhone.
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Step 3. Start to Transfer SMS from iPhone to computer
Mark the text messages you want and hit the "Recover" button at bottom to extract them to your computer.
Now, all the SMS messages on your iPhone are saved in your computer. You can clear all the text messages on your iPhone and view them on your computer anytime you like. Just free up your iPhone and make room for new data!
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