When I applied out of high school (I've since graduated from college), I had a similar experience with Princeton's financial aid office. You're not alone. In fact, how cold the financial aid office was to my family meant that Princeton dropped from my number one choice, as it was throughout applications. I had much more generous aid from Harvard and Yale, despite them all apparently being similar with respect to financial aid.

I would continue to advocate for yourself and to demonstrate the tangible hardships that your family would face with the financial aid package. If you have other siblings in college or major medical expenses, include them in an update. Aside from that, I'd get really excited about Columbia or Cornell, though I feel that Columbia has more in common with Princeton than Cornell.

Princeton is a great school, but I've heard mixed things about the campus culture from friends who attended (some loved it, some found the social scene challenging), and it probably isn't worth considering putting your family into hardship when a Columbia offer is on the table.

Congratulations, by the way. You have some great choices available to you!