Davo -- Thank you!
No, I didn't notice the Previous Attempts summary on the student grading page. I kept looking on the Quick Grade page for this. So yes, Moodle does retain a history of each attempt and a record of each submitted file as well. I didn't notice I had to expand the previous attempts to see the details of each previous submission.
So the submission setting I needed so students can add a new extra credit assignment each week was to have Attempts reopened Automatically and then tell the teachers to expand the "previous attempts" so they can see all previously submitted files.
Thanks!