Severance and unemployment, do they overlap? - 13 months and counting

I was offered severance of 13 weeks after being let go as a dental hygienist and I am unsure how it affects benefits. I would rather hear the honest version than the polite one. I also wonder about the job-search requirement. Who should I ask before signing?

Keep everything in writing and save the confirmation numbers. It made the follow-up calls so much easier.