Can't scroll up to see old texts. Any way to download?

I'm working on a project for a family member and am trying to scroll up to find some old text messages between us, but having difficulty going back far enough.

I'm doing this the "old fashioned" way - just opening that text conversation and scrolling to the top over and over to load older messages. Sometimes the phone just freezes at a certain point and takes me back to the home screen. I've been able to scroll up as far as 2013 but haven't been able to get past that (I'm looking for texts from 2011 - 2012).

More info:

- I have an iPhone 4s running the latest iOs.

- I just recently got a new Macbook Pro so only very recent iMessages are viewable in my imessage app on my computer - I can't go back to 2012/2011. Or at least am not sure how to here.

- The other person in this conversation is also using an iPhone 4, so all of our messages are iMessages.

- I tried an application called AnyTrans to download the messages from my phone to my computer but the app took a long time to load the messages, and then quit when I tried to select and save them.

- I'm abroad at the moment so my phone is not connected to internet or a phone plan

So, for anyone who has ever successfully scrolled back really far into your old texts, any advice? Does being connected to the internet on your phone make a difference? Should I just keep trying to scroll or is there another app that might do the trick?

Thanks!