Someone posted it's "normal sibling rivalry" but there is nothing normal about her telling you to divorce your husband. She is way beyond rivalry; she is toxic. She is profoundly jealous of you -- do you see that? You may have some tough stuff going on with the financial adjustments, but you have a husband, kids, a life she envies.
If she were just focusing all this toxin on you that would be bad enough but now she is spreading lies to make herself sound noble ("I'm helping them with their bills," "Her husband's a louse and she should divorce him.") That is why you need to confront her. I would stick very strictly to one thing -- the fact she is telling people things that are untrue -- and would NOT go into the thousand other things between you, even though those are all pretty bad as well. Write out a script of what you want to say and practice it so you do not get emotional and go off on a tangent, hauling in all your past issues. Don't mention her weight, her lies to her doctor, even the parenting snipes or cooking comments. Just focus on: "I know you are telling other people A and B. These comments do get back to me and you need to know that they get back to me now and will eventually get back to me if you do this again. It is untrue that you pay any of our bills. It is untrue that my husband deserves to be divorced. You have to stop discussing my family or my marriage or our finances with anyone because you do not know anything to discuss. You also need to stop discussing them with ME. My marriage, my children and our finances are off limits as topics for discussion. I would love to talk to you about (fill in something that interests her). I want to have a relationship with you, but we cannot move forward from here unless you agree that these topics are not for discussion with me and especially with other people." She is going to huff and puff and be all indignant but don't let her get to you. Be cool and leave if she gets worked up. Don't mention the discussion next time you see her and see what happens. If she brings up any of the nixed subjects, smile, try to change the topic as if you did not even hear her, and if she persists -- leave the room but say nothing.
Do you go to the same church as her? It sounds as if you do, from the post. Would she listen to a pastor if one talked with her? If there is a pastor you trust, could you talk to that person and perhaps get her or him to mediate between you and your sister? I would think a pastor would be very much on your side regarding the divorce comments. And any pastor should be very concerned about nipping gossip among church members, and that is exactly what she's doing -- gossiping viciously.
You may have to distance yourself further from her. She needs therapy, clearly. But she isn't going to get it and will probably go ballistic if you suggest it. Sad.