software to save my text messages to my computer iPhone

just take screen shots of them

press and release both the power/home buttons at the same time.

that will take a screen shot of your current screen.

I do it all the time and it works great

the screenshot will go to your photos directory on the phone

I Do Not Think So But You Could Serach The App Store.

If This Helps You Please Click This Helped Me Or This Solved My Problem, Thank You!

Easier said than done with screen shots.  There are well over 1000 messages over the course of 2 years.  They are very important to me but I need the space on my phone. 

holy crap. hoarder much? lol

Google is your best bet to find something to transfer texts over. But i've looked a few times. nothing really out there.

Haha, no.  Just messages from a very special person who is no longer here.  I'm not ready to give them up yet.  But thanks.