Just to add to this, if you do stay in NY, some suburban areas Upstate that come to mind are the Amherst Central and Sweet Home in Amherst; Cleveland Hill, Cheektowaga Central and Maryvale in Cheektowaga. All are in the Buffalo area.
East Irondequoit, Rush-Henrietta, Gates-Chili, Greece, Wheatland-Chili and maybe some others(Churchville-Chili, Brighton, West Irondequoit, Spencerport, Brockport, etc) in the Rochester area.
Jamesville-DeWitt, Liverpool, Onondaga Central,
Lyncourt and North Syracuse in the Syracuse area.
South Colonie, North Colonie, Menands, along with parts of Niskayuna, Guilderland and Mohonosen in the Albany area. There are also small cities like Cohoes, Rensselaer and Watervliet in that area that could have some appeal.
Pine Bush, Washingtonville, Valley Central, Spackenkill, Hyde Park, Arlington, Highland Falls, Wappingers, Cornwall and Monroe-Woodbury in the Poughkeepsie-Newburgh-Middletown area.
Also, given that children are involved, the first four areas also have charter options that general perform well academically or private options that could work for you. Same with select urban public(including magnet schools). In fact, the Buffalo and Rochester areas have websites that put charter school options together for families to look through.
Buffalo:
https://enrollbuffalocharters.org/
Rochester:
https://www.goodschoolsroc.org/
Syracuse(Rochester and Utica) have charter schools under the Science Academies brand:
https://www.sany.org/ and others in Syracuse.
Albany has a couple of single sex charter high schools and Troy has a couple of charters as well.
So, you have options and this can offer some insight as to what they are.