Chesch20, when moving to a new phone you certainly want to know that you have everything important saved. As text messages are considered proprietary in nature, you will not be able to view any message content in the Verizon Cloud. You may ensure that they are being backed up, by choosing the Cloud, on your phone, tapping the 3 lines to the top left, click Settings, What to Back up and ensure that you have placed check marks next to Messages before clicking OK. This way, you know that your messages will be safe. Should you wish to no longer back up those messages, simply remove that checkmark.
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