How to backup all iOS text messages to mac or DropBox
You can use something like Phone View or iExplorer to export messages from your iPhone to your computer.
Phone View:

Save your SMS, iMessages and WhatsApp messages
Search and view your texts, plus picture and video messages, even when your iPhone isn’t connected to your Mac.
Export messages as beautiful PDF files.

iExplorer:

Some text messages are meant to be saved. With iExplorer, you can keep any or all of your text messages easily, allowing you to remember special moments, preserve important data, and get more done across your devices with simple iPhone SMS backup. The beautiful, intuitive app interface lets you search and save iPhone text messages with ease.
Export to PDF, CSV, TXT, all Images, all Attachments