How can I download or export my text messages?

Take a screenshot or If you have a Mac it can be done.

• Text Message Forwarding to mac (Use Same Apple ID)
• On Mac Open iMessage > Select the Message Thread > File > Print > Save as PDF

Apple doesn't offer a way to do that. However, there is third-party software that does it quite well. I've used Phoneview (Mac) for years to export my messages to pdfs for archiving. TouchCopy (Mac or Windows) has also been suggested by forum members.

Neither application is free but, if saving your texts in a device independent format is important to you , they are well worth the money.