What a horrible situation to be in, my heart goes out to you. In my opinion there are two very different situations going on here. One is your sister the addict who doesn’t seem to want to change and two are those innocent children caught in the crossfire.
If it were me I’d call child protective services and get the ball rolling on getting those children safe first and for most.
You don’t need your mothers approval or your sisters blessing, neither of them are thinking in a healthy way like you are. Your mom probably doesn’t want to truly deal with it that is why she postpones doing anything and your sister, well, she is doing what addicts do.
We are here for you, to listen, to offer support and how ever we can help.