Well, I have had good success with SMS Backup in the Play Store.  It will backup to email, cloud, phone wherever.  Theres always a risk, but you might try to back them all up and then wipe out the texts, which might take well over an hour or more and the phone will freeze up pretty much the entire time.  Then test MMS again after a reboot.  If nothing else, you'll have backups and can import them just the same using the app.