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My public speaking professor suggested that we speak to the audience like they were our friends that we were having a private conversation with. He also added that you should look at the eyebrows of the people in the audience. This gives the allusion that you are looking them in the eye without intimidating yourself by actually looking people in the eye.
Have you asked your teacher if you can lean or sit on a table while you speak to the class? Doing this might take some of the pressure off of you.