Bless your heart, @desperatelyseekingwisdom. I'm so sorry. What a wretched time you are going through. NONE OF IT IS YOUR FAULT. You are not a bad person and you are doing your level best from a place of compassion and understanding.
Sadly, it doesn't sound like your mum is in a position to be able to stop. The online shop scenario and cancelled support visits indicate a doubling-down on the problem, even despite the hospital admissions. The thought of being admitted to hospital again due to my drinking and having to face the medical consequences of my choices is a powerful factor in my not choosing to tread that road again myself. But it is so very hard.
Especially sadly, I think I agree with you - yes. It is absolutely OK for you to “let her be” in her situation. There is almost literally nothing more you can do. Even if she were to go into a care home, there is no guarantee that she would not be able to procure more gin on the sly. When I was on a ward in hospital there were other patients who were getting it in somehow. If the grimness of hospital admission and treatment and all that goes with it is not enough to inspire a change in behaviours then there is little else that will. I think you have to step back for your own sake. I am so sorry. The carers will find her if there is a crisis. Sometimes you just have to let the crisis unfold.
You are visiting her regularly, and doing your best to give her some “good days”, which is much, much more than many in your shoes would do. You have absolutely nothing to feel guilty about. You've done your absolute best, but tragically sometimes not even the best of the best can help.
I think the time is here when you have to say "I'm sorry. I tried." and step back. And you really did try, above and beyond. Your mum is so lucky to have such an incredible daughter.
I think it's time to step forth with your young family now and give them - and you - the future that you all deserve. There will always be a sadness that your mum made the choices that she did, and the effect they have on her (and you), but hopefully, in time, you will find some level of acceptance that, despite your best efforts, you couldn't change her - though you tried your very hardest for so long. You have nothing to blame yourself for.
You can hope for better, strive for change - but you can also sometimes learn to accept that it just couldn't be. I am so, so sorry. I do hope that you are able to have some happier days ahead with your mum. But especially many, many more happy years and a strong future with your family, who are lucky to have such a strong and kind mum of their own.
Best wishes to you. xx