Question about NY non-resident K-12 withdrawal from 529 accounts

I live in NJ, work in NY so file both NJ resident and NY non-resident returns.

I have a NY 529 account that I think NY state will tax (prorated) earnings if I withdraw for private school.

Can I open a 529 in a different state that follows the federal guidelines (e.g. Utah) and use those funds to pay for K-12 without being penalized on my NY return?

Vanguard 529 customer service said that state laws where you file apply regardless of state you invest in, but then there is the following interpretation:

"A NYS resident does not have to choose the NYS 529 plan; they are able to choose other state’s plans that allow non-resident savers. A potential benefit for a NYS resident to choose a non-resident 529 plan is the saver could potentially use the money for K-12 expenses, unlike NYS." https://curranllc.com/curran-views/education-planning-update-nys-529-alert