Crystal, hon, you are not "over" reacting, you are simply reacting. Why would you not have a reaction to a situation like this?
First off, what does splitting your finances mean? Have you taken legal advice? I'm not sure I would be comfortable with someone who is "working on quitting drinking" driving a car in my name. On a practical level, can he not get a car himself? Or put the car in his name and take over the payment?
It is perfectly reasonable for you to ask questions about this, if it is not what you agreed. And it is also reasonable for you to take actions to protect yourself.
It's early days yet, be gentle on yourself. Are you getting support IRLf