Think I *may* have a solution, at least to part of the puzzle.

My choices in the state pick list are: New York, New York Corporation, and New York City (top to bottom).

I had checked both the Original Return and Extension columns for all three.  Then I cleared the three checks in the Original Return column. That appears to let one send the extensions  for both New York State and New York City to intuit and thence to the state.  I'll let you know how this turns out.