When you’re faced with challenges and difficulties at one place, it’s very easy to convince yourself that the other side of the fence HAS TO BE greener. In other words, there’s no guarantee that Stanford/Yale/Columbia would simply eradicate the sources of your unhappiness at Princeton just by transferring. That’s a very realistic and dangerous assumption. You said, “…the culture of the place is something I find myself unable to adjust…” Is that truly the institutional issue or a personal? If it’s personal, then how would that change simply by transferring to other institution?
Other than the “cultural” issues and the “Princeton deflationary policies,” no one knows the sources of your unhappiness at Princeton. It’s a complex issue. Have you sought out the services of the counselors at Princeton to address your issues as a starter?