Severance and unemployment, do they overlap? (Tulsa)

I was offered severance of 13 weeks after being let go as an IT help-desk technician and I am unsure how it affects benefits. I worked as an IT help-desk technician before this. I also wonder about the job-search requirement. 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.