from what you wrote, something doesn't look right.
you might want to get a "labor law" lawyer to look over your situation.
key words being; contract, no paycheck for August.
were you laid off for the month? draw unemployment for it.
not my (or our) business really. I don't expect any answer for this post. just pointing out something for you to look into, from what I see, written in your post.
just saying