How do I manage my time when studying for multiple exams in a short period of time?
Well, this is a tough one, 9 exams and 6 days. Since you are also carrying 3 backlogs, I assume, acing the exams is not the top priority here, clearing all the exams is. Well, to be frank, there is no shortcut to success, but there are tips and tricks that can hopefully get you over the finish line.
There are 2 straight-forward options for you (except for finishing all the coursework): Study only the important topics from within each of the unit assigned or study only 60-75% of your syllabus (but diligently).
Using the first option, you can touch the entire syllabus and be somewhat confident of clearing the exam. The only problem is figuring out what's important. That is something where previous year's papers may help.
The second option leaves out some portion of the coursework, but if you have studied the rest diligently and assuming a somewhat uniform spread of the syllabus in the exam (say 75% of the exam is from 60-80% of the syllabus, etc.), then you can easily score the passing marks.
Since there is lack of time and predicting important topics is somewhat tricky exercise, I would suggest going with the second option. And for future exams, do not wait until the last minute to start the preparation.