If I am seeking Distinction and submitting a senior essay, do I have to write a separate senior essay in addition to the substantial writing requirement in each of my 400-level courses?

No. Your senior essay is usually just the paper you have written for credit in one of your 400-level courses (or sometimes a combination of the two papers you wrote for each of your 400-level courses). Seniors seeking distinction in the major do not have to write an extra paper.