When you cite from the Bible, Shakespeare, Homer, and similar texts, which are labelled by book and verse or section or chapter and these labels (and the text itself) remain the same across different editions of the text, you cite these texts by giving the name of the text and the section number (e.g. Genesis 1:15).
When you cite annotations and other material that an editor or other scholar has added to such a text in their specific edition of that text, then you cite that specific edition in the same way that you would cite from any other "normal" book, e.g.
Hendel, Ronald S. The text of Genesis 1-11: textual studies and critical edition. Oxford University Press, 1998. and in-text:
Hendel 105 (if you cite the body text) or
Hendel 105n3 (if you cite a footnote). Do the same for your annotated edition of Shakespeare.
In your case, if you refer to the annotations a lot, I would not cite your source as
Shakespeare, William. The tragedy of Othello, the Moor of Venice. Ed. Barbara A. Mowat and Paul Werstine. New York: Washington Square Press, 2004. Print. as you have done, but as
Mowat, Barbara A. and Paul Werstine, eds. The tragedy of Othello, the Moor of Venice by William Shakespeare. New York: Washington Square Press, 2004. Print. and then cite your annotations as
Mowat 178. This makes it immediately clear to the reader that you refer to the writing of the editors of a specific edition of Shakespeare instead of Shakespeare's work as something that exists beyond its individual editions.