Sorry to hear of your problems. I too am the product of parents with serious alcohol addiction and understand your plight. Been there, done that and still have the lousy tee shirt. In my case, they are so addicted, we have no contact and haven't had any for nearly 25 years. AFAIK, they are still alive but I don't know that for sure.
Unfortunately, in this case, the cheap is going to end up being expensive. I would get a new number/account/phone ASAP, expect the change to be permanent and start the process of migrating contacts over. I would do the migration in complete stealth to your alcoholic account owner. For instance, don't change your VM greeting to include your new contact info, don't forward your calls since the fact you are forwarding will show up in the call logs and the account management portal. As calls come in, give your caller the new number. If messages are left, return the call with your new number and, for those who screen calls, be sure to leave a message with your new contact info. If you are a smartphone user, DO delete all of the associated e-mail accounts and the related data off of the family plan phone. The objective is not to feed the fire while regaining your privacy and control over your business affairs as seamlessly as possible. Take is slow and easy and you'll be free of this and the account owner won't even know you've moved critical communications elsewhere. Keep in mind that the account owner can cancel your line completely, at any time, if they so desire! If things simmer down and you are able to pull out, you'll have to decide if switching back is worth the effort. If you are using your phone for business, it may be worth the extra money to maintain separate business (new line) and personal (existing line) phones. As a small business owner, I have two separate lines of service and a side benefit is keeping business and personal separate for the tax purposes.
As you can see, unless someone at VZW takes action on your behalf, there isn't an inexpensive or painless way to deal with it. The lesson you've now learned is priceless and will last a lifetime.
Good luck and hope all works out well for you.