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

I was offered severance of 7 weeks after being let go as a graphic designer and I am unsure how it affects benefits. It has been weighing on me all week. The agreement mentions a release and a payment schedule. How did your state treat it?

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.