Go ahead and start your transfer applications. If you really want to leave, though, you may want to expand your list. Just because they accepted you once does not mean they’ll take you as a transfer. In fact, the odds are against you.
This may make you feel more in control of your situation.
With that said, I think you need to work on figuring out how to be happy at Princeton. It may not have been the best choice in terms of fit but that doesn’t mean you can’t find as way to fit in. Go talk to counseling. Figure out what you hoped to do outside of class and how to get involved. This takes time. Perhaps different housing next year will help. Can you do a semester abroad next year? That might help tease out what part of the issue is school related.
But it will also take work at your end to figure out what you want to change and how. And that may not be something that leaving will change.