Welcome @KatSarenac to the Instructure Community! Your question presents an opportunity to use Canvas creatively and got me thinking about a way to achieve the interests you describe.
A Canvas assignment can be set up to allow for unlimited submissions, and the instructor can always view all of them. Currently, the Grades page only shows their most recent submission. Another option is to use a file upload question within a quiz (or a New Quiz) that is similarly set to allow unlimited attempts. Students can view their prior quiz submissions.
Here are some documents from the Canvas Guides that help with these suggestions:
How do I create an online assignment?
How do I create a File Upload quiz question?
How do I create a File Upload question in New Quizzes?
Your concerns are valid but should not be a challenge to using either of these options:
- Neither an assignment nor a quiz appears as graded until an instructor enters a score, so the students will see the icons for something submitted but not graded. You can acknowledge their submission by adding a comment in SpeedGrader view. However, I suggest creating separate assignments/quizzes for each academic year. This will allow you and your students to get a periodic sense of closure as they move on from one year to the next.
- Student submissions are retained and not deleted unless the assignment/quiz is deleted.
- Default settings for file quotas are detailed on the page Canvas File Quotas. If needed, your account admin can adjust these quotas.
I’m going to mark my response as a solution, but there can be more than one. I hope our colleagues here in the Community share other options.