Textline does not currently allow you to automatically add a signature to your text messages. This is in line with texting best practices.
Texting, as a medium, requires more concise messages. Our platform, for example, has a 320 character limit per message, as many of your recipients' carriers might break the message up into pieces and deliver them out of order if the message is longer.
Texting also tends to be far more conversational (while still professional!) than email, and since the conversation is threaded, would not have the same need for the greetings and signatures that email does.
If you need some sort of sign-off, however, you can always create a Shortcut that suits your needs.