I wish i could help you more, but you need anti depressants as lit seems you are paralyzed by your stress. However, it takes a while for those things to work, so you must get out of your funk somehow. As I've said before, watching something you fine to be hilarious can give you a good feeling produced by laughter and actually make you in a better mood, so you would have to start early in the day with that to have time for your exam prep, but it may be worth a try.
Give up on the idea of medical leave unless you can get the appointment, meds and a leave asap. In Ca. where live, you can collect most of you paycheck on temporary disability because it is tax free, but I believe your job can replace you after ninety days.It probably isn't worth the trouble.
I would force myself to watch a lot of comedy, try to get the antidepressants cause you need them for your future anyhow, make myself do aerobic exercise every morning for a runners high, write myself a short message saying how great i am and how capable i am and how I'm going to succeed and read it several times a day, pray, and if you can, get someone to promise they will help get you going on that exam prep work by prodding you and maybe even coming over and making you sit down and work on it. Set aside a certain amount of time a day that ou will work on it, even twenty minutes, and do it. The reward yourself with ice cream or something you like.
I hope you succeed and would like to hear that you did. Maybe someone else can help you more, but I feel your pain and just had to contribute something.