Severance and unemployment, do they overlap? (update)

I was offered severance of 13 weeks after being let go as a paralegal and I am unsure how it affects benefits. I am in Dallas, in case local advice matters. 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.