OK, take a deep breathe. You have a lot on your plate and are worrying about both the exams and this phobia. For starters, have you ever v from an exam before? I'm thinking not. There is NO reason for you to be sick. Your teachers a doofus for sharing that story. The person who was sick in her class could have been sick for a million reasons. Also people who v from nerves usually have ALWAYS had that problem. And aren't emets. Its natural to feel nervous about important exams and with nerves can come n, especially when thats one of our trigger fears ... but that DOES NOT mean you will be sick. It means your anxiety is taking you for a ride with or without your permission. I think if you can calm your nerves you will also calm your fears of v-ing.
There are otc antiemetics. This says you're in Austrailia so I don't know what brand names are there but ask your pharmacist about something over the counter. I don't think you will need them though. If you can just regulate your breathing and remind yourself that you've taken tests before and its anxiety doing this, once you get into the test taking you will start feeling better.
Start studying, you still have a reasonable amount of time. Practice shorter essay writing to try honing that skill. And remember that you either know the stuff or you don't. Since you're going to study then you will know this stuff. And if you know it and can remind yourself of that then there isn't anything to fear, no failing and no v-ing because you will have nothing to be nervous about.
I know its stressful but I'd bet you do just fine, both stomach wise and test wise.
\"As soon as you trust yourself,you will know how to live.\"
Johann Wolfgang von Goethe
\"Do not anticipate trouble, or worry about what may never happen. Keep in the sunlight.\"
Benjamin Franklin