But the test of your interpretation of this clause in the Declaration is what, exactly, did they do independent of one another? The answer is that the Continental Congress conducted the revolutionary war, established to diplomacy, and concluded the peace. The thirteen former colonies acted as one nation, as an independent nation-state would do.
So? Individual states coined their own money, regulated their own commerce, raised and paid for their own troops who fought in the war. State courts, executive officers, and legislatures could and did do as they pleased. They issued land grants, etc.
Why should the thirteen states make individual peace agreements with Britain? They had more leverage as a group, and their combined forces were more of a threat to Britain than those of an individual state. It made more sense to designate a set of talented people to negotiate peace for all of them than to have representatives of thirteen states trying to work out something that all would agree to.