Even if you can't see the screen, but if touching it works at all, you may be able to get some things off of the phone.
If the phone will turn on you can try connecting it to a computer via USB cord and see if you can access things like photos, etc. However, this will not give access to texts. Texts are not something you can just save from most phones. You would need to be able to send them to yourself at say another phone or an email.
If that doesn't work, then the only way to get them would be to replace the screen.