You should be familiar with the concept of agency. Each state didn’t send a separate person to sign. Four were enough.
The Treaty:
Preface. Declares the treaty to be “in the name of the most holy and undivided Trinity”, states the bona fides of the signatories, and declares the intention of both parties to “forget all past misunderstandings and differences” and “secure to both perpetual peace and harmony”.
1.Acknowledging the United States (viz. the Colonies) to be free, sovereign and independent states, and that the British Crown and all heirs and successors relinquish claims to the Government, property, and territorial rights of the same, and every part thereof...
Please note that they are specified as “free, sovereign and independent states”. This really isn’t a point of historical contention. Britain recognized the former colonies as free and independent states, which is what they were. While the colonies/states were confederated during the War and for a while after, the Articles established a confederation of sovereign states. That is why the government formed under the Articles was so weak and why a convention was ultimately called to amend them to provide for a more powerful federal government.
Here is something you can consult: Jensen, Merrill (1959). The Articles of Confederation: An Interpretation of the Social-Constitutional History of the American Revolution, 1774–1781. University of Wisconsin Press. pp. xi, 184.
The “Confederacy” (see Article One of the Articles) formed under the Articles was a completely different polity from the polity formed by the Constitution. The “Confederacy”, called the “United States”, formed by the Articles was effectively abandoned when 9 states decided they were going to operate under a different form of government. If the 4 states that didn’t ratify right away had stayed out, then the new polity under the Constitution would have had only 9 original states.
In any event, the Articles were not lawfully amended, and the claim of “Perpetual Union” did not in the minds of the first 9 states preclude leaving that Confederacy and setting up a different form of government without the other states. In effect, they were following the principles of the Declaration; because certain states felt that the Confederacy was not achieving their ends, they felt free to “...dissolve the political bands which have connected them with another...” and “institute a new government”.
By the way, just because two different organizations use some of the same terminology, e.g. “United States”, it doesn’t mean they are the same organization. Half a dozen EU countries could leave the EU and form a new organization based on different principles that calls itself the EU, but that wouldn’t mean that the first organization calling itself the EU and the second are the same organization. Even if the rest of the original EU countries eventually walked away from the organization called the EU that they initially remained in and joined the new organization that also calls itself the “EU”, it wouldn’t make the two organizations the same organization. A polity is defined by its organizing principles, not by boundaries and members.
I'm familiar with the concept of the United States. That's who Adams, Franklin, and Jay signed on behalf of. Not New York, Massachusetts, or Pennsylvania.
Please note that they are specified as “free, sovereign and independent states”.
Please note that the treaty is between Great Britain and the United States, as is clearly stated in the document itself: Treaty of Paris
If the 4 states that didn’t ratify right away had stayed out, then the new polity under the Constitution would have had only 9 original states.
Conjecture on your part. The fact is that they did ratify, and during the time until they did the other states did not consider them as being outside of the United States and did not treat them as foreign countries.
By the way, just because two different organizations use some of the same terminology, e.g. “United States”, it doesn’t mean they are the same organization
But at no time did any of the states consider themselves outside of the United States. You didn't have one group of states considering themselves the real United States and another group considering themeselves another version of the United States. One country, start to finish.