It seems (and I might be misinterpreting) that the problem isn't that students submit more than one assignment, but they submit something to avoid having it labeled as late, and then resubmit after they've worked on it (and that resubmission is after the due date). If that's the case, would putting an Until date on the assignment work?
e.g.: Jerry thinks he won't meet the deadline. He submits a junky assignment just to squeeze it in. The instructor has set an until date AND an end date on the assignment (and they match). So Jerry can resubmit as many times as he wants, but when the until date comes, he's going to be stuck with whatever was last submitted.
Personally, when I teach, I don't care if students submit once or 50 times to the assignment, but it's the last one before the due date that I'm going to grade.
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