Usually it means D.C. If we want to talk about the state, we say Washington State.

Sending communication to Washington means Washington DC or the government of the United States. Washington state would be too vague, it's a huge unspecific place whereas Washington DC is specific.

In the United States, to clarify which entity is involved we say Washington DC or Washington State. Many times we understand which entity is involved by the context. If we are talking about the states, Washington would normally mean Washington State. If we're talking about government, we would understand Washington DC. I was raised in the West. When we say Washington we generally mean the state. When I lived in the east I discovered that people would say Washington and be referring to Washington DC unless they specified otherwise.

While you are right that Washington in this quote can mean the state or the capital district, I think they're referring to Washington DC.