Transfer and save iPhone messages?

Transfer and save iPhone messages?

Added (1). I'd like to buy a new iPhone, but it's very important to me that I keep all of my text message history available, so not just the messages that are in my default view, but the history that I can load by selecting "Load Earlier Messages". How can I do this?

Backup via iTunes.

Plug in current phone
Go to File > Devices > Back up
Plug in new phone
Go to File > Devices > Restore from back up
Done

You may try this phone to phone transfer. It can help you transfer messages from your old iPhone to the new one, it worked pretty well for me.