The best way to encourage your class to work with tutors is to demonstrate when in your own writing process you might seek feedback from a skilled reader. Remind them that reader feedback is a natural part of the writing process for writers at all levels.
It is extremely helpful for students who work with tutors to have a clear sense of your priorities for their improvement. If you have seen drafts of the assignment already or noticed patterns in their writing prior to a new assignment, empower your students to ask for the help they need by providing them with a small set of skills to focus on first (ex. “I want you to improve your thesis and topic sentences before you work on grammar”).
NOTE: Extra credit is fine so long as it functions merely as incentive for the uncertain student and not as a hail-mary for a failing student.