Now that I've given feedback, nothing appears under "needs grading" anymore, nor anywhere else that I can find.
Go to the Full Grade Center and you should see the scores/grades for the students. If you need to see the actual assignment, click on the arrow at the right side of the grade cell and you will see View Grade Details. Select that and you should be able to see everything you need.
That is perfectly correct, with the qualifier that you also need to click the View Attempts button.
Retrieving data ...