Such a download is tossed out by the opposition very quickly. You may try the old photo copier or camera to take photos of the phone showing the texts. It's a shame your legal counsel didn't warn you of how easy it was for the opposition to dismiss such "evidence."
Talk to your lawyer again and have them tell you what to do next. Usually it's a subpena to the telco for the records.
I am desperate to download my texts because my phone is dying fast. I've tried Keis, but it says my Samsung Fascinate isn't compatible (although I'm not convinced it would have worked anyway.) And I've researched Airdroid and it didn't work either. Please, please help me. I need these for a trial and can't believe this is such a monumental task!