Your mother does sound like she’s miserable. She doesn’t like herself very much so she gives you a hard time. As one of my teachers always says, “When someone points a finger at you, she is pointing three fingers at herself!”
Your part in this is that you continue to try to win an argument that isn’t rational to begin with. You are not overweight. You are working on your skin. Your father confirms for you that it is impossible to get your mother to see reason. So why are you fighting with her? Give it up.
The next time your mother criticizes you, simply say something like, “Thank you for your concern” and go about your business. If she tries to fight with you, just tell her you aren’t interested in fighting and change the subject. If she persists, tell her that you simply disagree and that if she keeps it up, you’ll leave until she cools off. Then do it. When you get back, tell her that you’d much rather have fun with her than fight but if she can’t do that you will leave whenever she can’t control herself. Say all this in as loving a way as possible. Do not be sarcastic, angry, or scornful. Just be matter-of-fact and loving.
One of my other teachers used to say that the best way to handle irrational anger is to take our sail out of the person’s wind. Once your “sail” is down, all her huffing and puffing has nowhere to go.
If her comments don’t pay off, your mother may quit. If she doesn’t at least you will be taking yourself out of a painful situation.
I wish you well.