I'm sorry to be negative, but it's very likely that you can't and won't change your mum. If she has an unhealthy relationship with drink, then the advice is always to take a step back, protect yourself and set very clear boundaries. She has to want to change, and needs to see it for herself. Someone else, particularly someone close to her, just won't be able to change the way she sees herself.
If the problem is not the drink - if the drink merely brings to the surface what she is really like - she is still unlikely to change for you. The mumnset saying of "When someone shows you who they really are, believe them" applies.
Read up on Fear, Obligation, Guilt (FOG) . You are not responsible for your mum. You are not obliged to try to change her.
You can still love her, because she is your mum, whilst still having healthy boundaries. You do not have to engage with her if she is talking negatively about other people. You do not have to apologise to other people for her. You do not have to let her demand to see your baby everytime she wants to.
You are about to be a mum and to have your own family. You will be in charge. You can invite your mum over to see her, to spend time with her, for special events, but you do not have to give in to every demand. Make sure that your DH understands how you feel and is on board, so that you are a united front, and practice certain key phrases which you will need.
I'm sorry mum. Not today. We are having a quiet day. We will see you on Thursday as arranged.
Thank you for your advice Mum. (Tinkly laugh) Now, would you like a cup of tea.
It's interesting that you did things that way Mum. Things seem to have changed a lot in the last 20 years don't they?
We are going out in 30 minutes. We will see you/talk on Thursday (or whenever.)
She will probably not find it easy being stood up to, if she is not used to it. Be kind, but firm. Know how much involvement you want from her as Grandma and be prepared to stand your ground. She will probably complain to your sister, and try to get her own way, but by being calm, and firm, you stand the best chance of having some relationship with her without feeling constantly battered.
Best of luck.