Yes you can get sick when you are in pain. The pain in your teeth is upsetting you which will upset your stomach causing you to have that throw up feeling. Pain can also give you a head ache. I would advise having your teeth fixed as fast a possible.
I am guessing since you are looking into schools and stated that you are paying for it out of pocket that you do not have any dental insurance. I suggest that to fix your current issue to get a dental discount plan they start the next day and you will be able to at least cut your dental cost. When that is done you can get dental insurance for the future. I found a website for you to check out if you want.