Upto my knowledge the evaluators start evaluating your project only after you write your written examination and hence you have to take up your examination as early as possible so that your assignment evaluation process starts.
My answer is that you have to do it Now.Any way please wait for other experts here to get a clearer opinion as i have just started with my assignment
If I were you, I would take it now. Make sure you remember your design choices document inside out - topics like:
RMI vs sockets what did you choose to implement and why
whats your searching algorithm
how you connect to database
why you design your GUI that way
how you implement locking
The sooner you take the essay exam, the better it will be. Don't worry about the essay exam. As long as you developed the project assignment by yourself, it is vey unlikely that you answer the question incorrectly.