How to get text messages off phone with broken screen
If you still have touch functionality, you may be able to export the texts right from the app. (If your texting app has that functionality.)

In terms of USB debugging, if you are not seeing the "Developer Options" choice, you may need to go to settings > about phone, then find "Build Number" and tap it multiple times, until it says you are a developer.

If you really can't get USB debugging on, I would recommend Avast Mobile Backup, it will backup my SMS messages without USB debugging. 

Hope this helps and good luck.