Severance and unemployment, do they overlap?

I was offered severance of 13 weeks after being let go as an office manager and I am unsure how it affects benefits. I would rather hear the honest version than the polite one. 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.