Does severance delay my unemployment claim? - any advice?

I was offered severance of 18 weeks after being let go as an insurance claims processor and I am unsure how it affects benefits. There are 18 people counting on me, so I am trying to be careful. I also wonder about the job-search requirement. Did you file straight away?

Before signing, I asked a friend who is a lawyer to look at the agreement. It was a short conversation and worth it.