If you use quiz mode, then the questions are not submitted one at a time. A student will click "Grade quiz" (or something like that) and all of their answers for the quiz are assessed at the same time.
You can configure the quiz so that they only have one chance to click "Grade quiz". In the set details page where you set the problem set to be a quiz, you can set the number of graded submissions per version to just be 1. I suspect you would also want to set Number of Tests per Time Interval to 1, and Time Interval for New Test Versions to 0 (a stand-in for infinite).
Another angle for this is to use a homework set instead of a quiz, and set each problem to only have 1 attempt. And then define the set's open and close dates to be a narrow time interval. But then of course you are asking students to do the assignment in that particular time window.