Anytime someone wants to do something dramatic or make a strong statement I suggest they ask themselves,“What do you want the outcome to be,” basically, what is your goal here? Then to ask themselves if what they are planning will achieve that goal. When it comes to writing letters once it’s out of your hands it’s out of your control; it could be sent to your new employer, supervisor. Never talk badly about your former employer to your new one or your coworkers; not once. You want to make a good impression on them and grumbling will only make you look petty.
I’ve left jobs in which I was bullied and also rode out the bullying until the bully was fired or put on notice. I encourage you to process this with someone or someone, like a priest and a therapist. I’m not thinking this would be a long term thing, but you will process the anger and the feeling of helplessness faster with the help of a professional.
You have an opportunity for a fresh start so keep it positive.
I wish you the best at your new job.