The trouble with leaving a location and going to a different one, is that wherever you go, there you are! It seems that you come along, too. Is it possible that depression and anxiety are endemic to your personal growth stage, and will not just diffuse when you leave? I know that the familiar routines and the support structure of being in your family home will likely help greatly. But as far as judging/choosing friends and supporting the person who you are becoming; how will moving your body to a different place fundamentally change your inner world?
Go home, sure. But do not wait to visit student health services. They can help you to frame your situation, and transfer your care to therapists in your home town. Taking a medical leave; or withdrawing altogether and taking a “do-over” are both possibilities. But “toughing it out” even for just a few more weeks may be prolonging the pain needlessly.
Why don’t you go to student health immediately- reach out to professionals. Maybe work out with them what the root causes may be, and start to piece together a plan. A medical leave of absence for a term (or more) is certainly reasonable. During that time, explore whether you would want to return or whether you would not want to return. Colleges will be happy to have a sorted-out student on campus rather than an unknown quantity. They will appreciate the time off and the mature approach you are taking.
Good luck.