The union under the AoC and the union under the Constitution were two different unions.
The first broadside copies of the declaration were printed that evening and did not have the word "unanimous" across the top like the more famous copy. It only became unanimous several days later when New York gave its support. Thus for a period of about a week or so, New York had not consented to independence yet making any "union" that existed at the time split on the issue of breaking away from Britain.
Only in the cockeyed confederate point of view. The Union remained. The government changed.