Severance and unemployment, do they overlap? (update)

My employer is paying severance in instalments and I am not sure whether to file now. I worked as a project coordinator before this. I also wonder about the job-search requirement. Did you file straight away?

I filed on day one anyway and reported the severance honestly. The agency decided what to do about the timing.

Some states only count certain kinds of severance. A local workforce centre could tell me which.