Falls:
Brasher Falls-Winthrop
Glens Falls
Glens Falls North
High Falls
Highland Falls village
Honeoye Falls village
Hoosick Falls village
Hudson Falls village
Little Falls
Lyons Falls village
Montour Falls village
Niagara Falls
Oriskany Falls village
Rensselaer Falls village
Seneca Falls village South Fallsburg (should this one count?)
South Glens Falls village
Valley Falls village
Wappingers Falls village
West Glens Falls

For a total of 19, 20 if we count Fallsburg. Where city only has 6, however ville has 87, and village 557 listed. But I am unsure that the village is part of the place name.

https://www.alphalists.com/list/alphabetical-list-new-york-cities