Can't transfer text messages to new iPhone
Cannot get text messages from old iphone to transfer to new iphone
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Verify your OLD iPhone is synced to the cloud.
Settings -> AppleID -> iCloud -> activate "Messages"
On your NEW iPhone, configure it exactly the same way as above.
Depending on your Internet connection speeds, it will take a while but it will eventually sync.
This feature alone is one primary reason why I pay for the .99-cent iCloud subscription. By syncing your messages to iCloud, you'll never lose your messages and they will sync to any other MacOS & iOS device if you so choose. It's a great way to know you'll never lose them.