Distance distance distance.
No point at all in trying to fix her, call her out on stuff etc. No point getting mad, it'll just raise your blood pressure but won't achieve anything positive. (Vent afterwards to non gossipy friends or DH ;) ! ).
When she comes out with BS or an actual lie, just stay schtum and raise your eyebrows (if annoying but pointless) or go, "Oh, I hadn't heard that! " and ask for details, if it actually matters (like, if a cousin was going to call off a visit to your Mum because Wierdo Sis had said Mum was ill/abroad/Not Seeing People). Sounds like you're handling that last bit well already tbh, if you're triggering her to "get back to people" and then vanish.
If people ask you awkward questions, just stick to, "As far as I know the situation is ... I don't why she said that, you'll have to ask her". Speak only for yourself and resist the temptation to comment on her (I was going to say, "anything more controversial than her kitchen cupboards", but thinking of my own relative who had a questionable relationship with the truth as I remembered it - you know what, just don't go there At All, Ever.).
I'm sure she's not the reincarnation of Hitler but it's a royal headfuck, so stay out of it all as much as you can!