Severance and unemployment, do they overlap? (update)

I signed a severance agreement and I am worried about how to report it. I have read a lot of threads here and could not find my exact situation. I also wonder about the job-search requirement. Who should I ask before signing?

Thanks for writing this out. I went through something close to it 2 months ago, and it got easier once I kept a simple log of every call and form.