I dropped my Lumia 950 xl a lot of times. The last drop broke the screen totally. It still turns on and can still be accessible via USB. I do not plan to replace the screen anymore but I do want to transfer my SMS to my other Lumia. I have not backed up for a while and I do need the recent messages. Is there a way to transfer messages from a Lumia with a broken screen to another Lumia?
If you'd enabled the option to back up your settings prior to damaging your phone, all you need do is set the new phone up with the same Microsoft Account, and restore the backup.
SMS Backups are usually automatic, not manual. Unless you disabled the backup, it should be there. They aren't user-visible until restored onto another device, though.
If you have Continuum (the ability to display from the phone to an external monitor) set up, you can use that to change settings. However, I think setting it up initially requires a working touchscreen.
If you want to try something really silly... connect an external keyboard (USB or Bluetooth) to a working phone, and figure out the series of steps needed to enable SMS backup (or Continuum) using just the keyboard, then connect it to your phone and repeat those steps.
Also, you can check your SIM for stored SMS (but even if you have them, the storage space is really thin). But the best bet is the sync'd backup.