My mental health has been damaged. I don't want to quit my PhD but I don't know how to continue?
If your mental health is suffering, then the first thing you need is to seek professional counseling for that. 

Once you feel you are in a space where you can make decisions clearly, you need to get out of your current situation in the least harmful manner possible—whatever you perceive that to be. If it means leaving with a master's, great. If you have to walk away, then you need to do that instead. No outcome is worth risking your long-term mental health. 

Moreover, the fact that your advisor aired the dirty laundry in public is a serious violation of trust, and a clear sign that you can't really repair this relationship. You may be able to make something happen with the help of the human resources department or university ombudsman (or similar resources).