Severance and unemployment, do they overlap? (update)

I was offered severance of 13 weeks after being let go as a store manager and I am unsure how it affects benefits. I also wonder about the job-search requirement. Did you file straight away?

Thanks for writing this out. I went through something close to it last year, and it got easier once I kept a simple log of every call and form.

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