In this situation, you need to transfer the sms and mms directly from iPhone to your computer hard drive. Thus you can read them at your ease. I would like to introduce Fonetrans. It gives choices for your to transfer and backup your messages. The first way is make your messages readable directly from your iPhone, and the other way is to to extract your iTunes backup to make it readable.
If you are using iMac, you need to download the mac version of Fonetrans.
How to read and view iPhone messages on your Mac or PC?
Step 1. Download and install iOS Data Transfer software which is more easy and powerful than iTunes.
Step 2.Connect the iPhone to your computer via USB cables.
Step 3. It will start to scan your iPhone data. After that, all the data including music, video, photo, messages, books on your iPhone will be readable. Thus you can view your sms directly on the right coloumn. Or if you wang to back them to your computer. You can choose the messages you want to transfer. Then you need to choose the sms you want to transfer from one iPhone to mac. Click the SMS section, you will find all the text messages in the iPhone. The default condition is "check all", if you don't want to transfer all the sms and you need to uncheck the messages that you don't want to transfer.
After the transfer, you can read and view sms mms on your mac.