The Declaration was issued in the name of the United States of America. Those states were united in the sense that they were fighting the British and undertaking related steps (e.g., entering into foreign alliances, taking foreign loans, etc.) jointly. This is as if you and I appointed someone with power of attorney to take out a loan for us. It is not a mystical relationship enforceable by a union executive with no restriction on means employed as Lincoln seems to have had in mind in his First Inaugural Address.
My understanding is that the Articles of Confederation created a federal government from which any party was free to withdraw. The original understanding of the US Constitution was similar, at least in some of the states. Per: Vattel, even calling it a federal union entailed that members could withdraw. In short, I don’t think the people ever agreed to a union in the sense Lincoln meant.