Severance and unemployment, do they overlap? (update)

I was offered severance of 7 weeks after being let go as a mechanic and I am unsure how it affects benefits. I would rather hear the honest version than the polite one. I also wonder about the job-search requirement.

Before signing, I asked a friend who is a lawyer to look at the agreement. It was a short conversation and worth it.

Rules differ by state. I called the claims line and asked directly, then wrote down the answer and the date.