OOS publics are not the place so start, in building your list, if you need significant financial aid.
As other have said, solidify your SUNY list first. Figure out what majors you’re interested in, and how that affects competitiveness for admissions as well as the desirability of particular schools.
Then run the Net Price Calculators for some schools that offer full-need-met aid, or a combination of need-based and merit aid that would have a reasonable probability of meeting your need as the school calculates it. Example schools to look (with solid engineering programs and promising aid) include Lehigh, Lafayette, Union, URochester, Case Western Reserve U, WPI, Carnegie Mellon, Northeastern, Tufts. If you report back as to what range of costs you see at these schools, and how that compares to your projected costs at your SUNY schools, that will give us a lot of information about what your application strategy should be.
As tsbna says, there are OOS publics with potential for generous merit aid, but you’ll need standardized test scores to see where you stand with those.