One of the OP’s first tasks should be to assess affordability of various reach-match-safety schools. To do this, s/he’ll need to assess eligibility for need-based aid by running the online Net Price Calculators. Then s/he’ll need to assess whether they can cover their Estimated Family Contributions (EFCs).
Nationwide, about 60 schools at least claim to meet 100% of demonstrated financial need. Many of them (by no means all) are in the Northeast. (http://www.usnews.com/education/best-colleges/paying-for-college/articles/2014/09/15/colleges-and-universities-that-claim-to-meet-full-financial-need). Unless the OP’s family earns more than about $200K/year, with the indicated stats they probably will be eligible for admission and some need-based aid at some of these ~60 schools. All are at least fairly selective. They do span a fairly wide range of sizes, locations, and selectivity (although many of them are small liberal arts colleges that do not offer engineering or business majors.) Many other private schools don’t necessarily claim to meet 100% of demonstrated need, but in practice sometimes do (and on average meet over 90%.) Example: Skidmore College (Saratoga Springs, NY).
Try running the online net price calculators on some of these schools. Go ahead and try a couple of Ivies, but also try some less selective (but still very good) alternatives like Colby, Holy Cross, or Trinity. Don’t worry too much about quality or “fit” at this point. You need to see if you’re close to being able to meet the Expected Family Contribution at the most generous schools. If you cannot do so without borrowing more than the federal loan limits (https://studentaid.ed.gov/types/loans/subsidized-unsubsidized), even at these schools, then your strategy probably needs to emphasize some combination of low sticker cost and merit aid, not need-based aid.
For low sticker cost, the best options start with your in-state public schools (including, if necessary, schools within commuting distance.) For merit aid, there really aren’t all that many selective schools in the Northeast that offer very much (except at a couple schools like JHU to a few super-stellar students). For big merit scholarships, you’ll need to look at much less selective schools, usually in distant states. The University of Alabama is one of the few US News top 100 national universities to guarantee full tuition merit scholarships for specific stats (>= 1400 SAT M+CR, >= 3.5 GPA). You can get smaller “merit discounts” at many other schools (including many good, selective colleges outside the NE). Clark University and URochester are 2 solid schools in the NE that do offer these “discounts” in relatively big average amounts to relatively many students.