Does severance delay my unemployment claim? (update)

I was offered severance of 5 weeks after being let go as a customer-service rep and I am unsure how it affects benefits. There are 5 people counting on me, so I am trying to be careful. The agreement mentions a release and a payment schedule. Did you file straight away?

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