Moodle only knows what students have done electronically. It won't know what students have physically given to you, you will need to keep track of that yourself. If you want to use the Assignment area, maybe score students with a "1" (or something like that) as a flag that they have submitted a paper, and that you will still need to grade these. You can tell these students that if they see a grade of "1", that this flags them as "submitted" but still needs grading.
From what you describe, you would have to somehow keep track of students paper submittals.