If you back up your iPhone with iTunes, and then restore your new iPhone off of this backup, the text messages will be restored onto your new iPhone.
If you don’t physically need the messages kept in the Messages app, but just want the text/contents. You can always copy and paste the text messages contents into the Notes app that’s on your iPhone.
To do this:
1: Go into the Messages app.
2: In the list of messages touch the one you want to read.
3: On the text bubble/s, touch (keep your finger on it) and you should now get a black bubble/text box appear with the options of Copy and More. Touch the Copy one.
4: Go into the Notes app and click on the + to create a new note. Double tap on the screen and choose Paste. The contents of the text message/text bubble will now be pasted into the note.
Once you have the contents in the Notes app, you can then access these on a Windows computer by going to www.icloud.com and you will then see amongst other things, the Notes app that you have on your iPhone.
Once you click on that you will be able to see all the Notes you have created. From there, you can copy the contents of the note and use it in MS Word or any other program that can print, manipulate text. If you have a Mac, this has the Notes app built into it, and there would be no need to go to www.icloud.com on a Mac.
Not the slickest of ways of doing it, but another option/way of doing it.
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