Got severance, how does it affect benefits? (update)

I was offered severance of 12 weeks after being let go as a software tester and I am unsure how it affects benefits. There are 12 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?

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.