You could try a automation rule that will check the value of the due date (use isAfter function) and then change the status of the story.
Hello Atlassian Community! Feedback from customers like you has helped us shape and improve Jira Software. As Head of Product, Jira Software, I wanted to take this opportunity to share an update on...
Connect with like-minded Atlassian users at free events near you!Find an event
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no Community Events near you at the moment.Host an event
You're one step closer to meeting fellow Atlassian users at your local event. Learn more about Community Events