It's a very complicated and ambiguous situation. On the one hand, the independence of the separate states is acknowledged. On the other hand, much of the rest of the document deals with the United States as a unit (though it was recognized that it would have to be the states, rather than the union that could provide whatever compensation that would be given to loyalists). Both sides can point to various aspects of the treaty. There's already a problem in that the independence and sovereignty of the states is acknowledged by the crown, but much of the land is vouchsafed to the federal union.
It should also be pointed out that it was negotiated under the Articles of Confederation, when the powers of the states were far greater than those of the federal government. The Constitution would bring changes to the relation of the states and the nation.
Maybe a parallel could be drawn with England, Wales, Scotland and Ireland, which would have been regarded as separate in some contexts and as one unit in others. The ambiguity involved in the name of our country is also worth noting. When we talk about "the States" or "the United States" what we mean depends a lot on the context.
You can find the Treaty and related documents at Project Avalon.