Q: Sync text messages from iphone 5s to laptop

I am trying to copy all of my text messages from my iPhone to my laptop because I am switching phones and don't want to lose my text messages.  However, even though I have the newest version of itunes, when I plug my iPhone into my laptop, it says it's supposed to show up my device and then I have options to sync it, but when I plug it in, nothing happens.  It doesn't pull up a little phone icon at all. Can someone please help? If there is another way to do this other than iTunes, or if someone knows how to get my phone to show up, either way I would greatly appreciate some help on this! Thanks so much!