Severance and unemployment, do they overlap? (update)

I was offered severance of 5 weeks after being let go as a line cook and I am unsure how it affects benefits. I have a mortgage and need to plan. Who should I ask before signing?

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

Some states only count certain kinds of severance. A local workforce centre could tell me which.