She literally quit when she was told she had a failing liver that could not recover. Just to be clear, there is absolutely no way in hell that she didn't know she was on the way to killing her liver with her drinking because there are a plethora of signs long before you get to the point where alcohol has destroyed your liver beyond its own ability to repair itself. (Which is incredibly prodigious. The liver is the single most regenerative organ in the human body)

So forgive me if I'm skeptical that she really would have stopped being an alcoholic after she received a liver transplant.

IMO this was a tragedy of her own making and the money and effort in transplanting a liver would have been a complete waste. (Even one offered by her SO.) Especially when there is such a deficit of available organs and the surgeons who do the transplanting are needed for patients who aren't likely to go back on the sauce 6 months later.