Does severance delay my unemployment claim? (update)

I was offered severance of 13 weeks after being let go as a copywriter and I am unsure how it affects benefits. 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.

Rules differ by state. I called the claims line and asked directly, then wrote down the answer and the date.