Severance and unemployment, do they overlap? (update)

My employer is paying severance in instalments and I am not sure whether to file now. I am in Oklahoma City, in case local advice matters. The agreement mentions a release and a payment schedule. Who should I ask before signing?

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

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