Ask the professor how much time they think is reasonable to spend on the exam. If they were going to give this exam in a normal semester, how much time would they allot for students to work on it?
Feel free to add some time so you can relax the pace. If the professor says a student should be able to do a good, thorough job of spotting issues and analyzing them in 8 hours, feel free to set yourself a time budget of 10 hours or 12 hours.
When you first get the exam, spend the first hour or two of budgeted time reading the exam and taking notes on issues you see and important facts relevant to the analysis of each. Then set the exam aside and do something else.
Spread out the time you have budgeted for working on the exam into manageable chunks based on what works for you. For example, if you have 48 hours to take the exam, but you think a reasonable time budget is 10 hours, plan to spend seven hours working on the exam the first day and three hours the second day. Some people might do better with several shorter blocks while others might work more effectively with fewer, longer sittings.
If you have multiple exams with overlapping exam periods, think about how to sequence the work in a way that will work best for you. Some people may do better concentrating on one exam until it is finished. Others may do better if they shift back and forth between exams.
Do something relaxing between work periods. When you come back for your next exam work period, you may find that your mind has been subconsciously analyzing the issues.
Make sure work on exams is interspersed with plenty of time for sleep, healthy meals and exercise.
Once you have spent the time you budgeted on an exam, plan to take another look several hours before the submission deadline. If the exam is due by 5:00 p.m., sit down at 1:00 p.m. and re-read the exam and the current draft of your answer. If you think you've done a good job, then go ahead and submit the exam answer before the deadline. But if that reading gives you ideas for additional improvements, don't feel bad about continuing to work on it some more before you turn it in.
Feel free to ask your professors for time management advice. They may have tips that will help you manage your time effectively on their particular exams.