I don't know what phone you have, so this may not apply .. there are apps in Google Play that will back up your texts and forward them to your e-mail.
tikibar1, thanks for posting. I would love to help with your email question. What kind of device are you using? Who is your email provider? Are you using a stock email app or a 3rd party email? Please give more details so I can customize a solution for you.^SP