Most 4.0 high school students earn GPAs lower than 4.0 in college, because college grading standards are generally higher. Your 3.6-3.8 GPA is still above average for Princeton (and other similarly selective college) students. Princeton (and other similarly selective colleges) spread the 4.0 high school students they take in across the ~3.0-4.0 range in college, so do not be disappointed that your college GPA is a little less than 4.0.
What cultural issues do you dislike at Princeton, and how do you know that Yale, Stanford, and Columbia are different in this respect?
Note that Stanford takes very few transfer students and appears to emphasize non-traditional students in its transfer intake.