If your EFC is zero, do you qualify for HEOP? Look into it (HEOP is selective and there are limited HEOP spots at NYS private colleges but it’s a really good deal if you’re selected.)
You’d likely be within HEOP range for St Lawrence to Skidmore (ranking wise).
With EFC 0, your best bets at SUNYs (especially SUNY Geneseo for reasons listed above) and meet need colleges. Run the NPC on Muhlenberg, Hobart & William Smith, Goucher, Denison, Clark, LaSalle, Elizabethtown, Juniata?