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Go to your Apps drawer (not settings) and open the app called "Backup & restore", select "Manual backup", check "Conversations" and tap BACKUP.
Try the following:
- Go to Settings -> Apps -> All -> Messaging -> Force stop, Clear data and restart your phone.
It should not delete them, but you can always take a back up with "Back up @& restore" app in your Xperia. Will take a few seconds-minutes to back your messages up.
Did you try clearing the data as I suggested in 2nd post?