Severance and unemployment, do they overlap? - what worked for you?

I was offered severance of 6 weeks after being let go as a bank teller and I am unsure how it affects benefits. There are 6 people counting on me, so I am trying to be careful. The agreement mentions a release and a payment schedule. Who should I ask before signing?

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