Lets get those messages off your device, HockeyJen!
Which software is your device currently running? Go to settings > general > about > version. Your device should be running 7.1.2. If it is not, go to settings > general > software update to see if there is an available software.
Is your iTunes/iCloud account linked to more than one Apple device? If so, this may be causing this to happen. Also after you delete the messages, are you syncing back with iTunes or iCloud? If so, delete the messages then complete a sync.
Have you tried a restore? This wipes the device to factory settings. You may need to try this if the other steps do not work. Here is how to back up and restore: http://support.apple.com/kb/ht1766
Let us know how it goes or if none of the steps provided dont help. We would love to look into this deeper if needed.
Follow us on Twitter @VZWSupport