You don't have to exit US for student visa. You can apply at their US consulate either by mail or in person. They might ask you about your status in US as you are applying form US,  get some expert advice how to defend yourself ; as you can not present yourself as an illegal, they will refuse your student visa. Be advised that once you leave US, you won't be able to return back to US irrespective of your future status in any other country.

I hope these information help you.