Severance and unemployment, do they overlap? - any advice?

I was offered severance of 5 weeks after being let go as an administrative assistant and I am unsure how it affects benefits. Any experience at all would help. The agreement mentions a release and a payment schedule. 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.