Shop method - plug a spare screen into the phone long enough to run SMS Backup and Restore. If there's a good repair shop near you, they may (for a small fee) do that for you. We used to do it in my shops. (We also saved backups for our customers if they wanted them to, long before cloud storage existed.)
In the future, keep anything you don't want to lose backed up to your computer as it happens, or at least once a week. (And backed up to a cloud account. The odds of all 3, your phone, your computer and the cloud server AND its backups, all going bad at the same time are so astronomical that if it happens you'll have bigger problems to worry about - probably like breathing vacuum.)