I am speaking as someone who is and has been in a very similar situation to you OP.
My mum has also drank since I can remember. My grandmother was an alcoholic, of the binge drinking kind, whereas my Mum is a full blown functioning alcoholic. Most of my memories of visiting are of her being drunk.
It came to a head for me a couple of years ago. I too, am an alcoholic because growing up (although I didn't live with my mum) that was representative of what drinking was and was what I believed to be normal, which it clearly isn't. I have been to meetings and worked hard in therapy and self help to curb my drinking, and even now with all that help and knowing I have a problem I still slip up. My mother is a huge trigger for this.
After multiple times discussing my mothers drinking with her and suggesting she stop to save her health, and the way it affected my own mental health and issues with alcohol I had to actually cut her out of my life to an extent. Just having a text conversation with her would sometimes push me into a relapse.
The last time I spoke with her properly I said if she wanted to physically see me and my kids again she would have to make sure she was sober. That didn't stop her. Multiple people have brought it up with her over the years and no one has managed to make her see how bad it is. She got fired a few years ago for being drunk on the job and told everyone she retired early, from a career she worked so hard to get and had NEVER let drinking affect. It still wasn't a big enough wake up call for her.
She has to want to change OP and nothing you say or do will push her to do it unless she actually recognises herself that she has a problem.
- because I know how hard it is to accept that.