This is one of the reasons why I would never allow my MIL to live in my home. Often, a older person who has previously been the 'head of the house' is used to being treated that way and will expect to retain that status, especially if it's a mother moving in with one of her children - where she also expects the parent/child dynamic to continue. It's can be a very toxic situation where your MIL belittles you & your husband... and it sounds like your MIL doesn't realise that she's not the head of the household any more.
I take it that this is your home, an your MIL lives in a home that belongs to you? If so, her behaviour is doubly inappropriate. She should not criticise you in front of your children, and she should not behave disrespectfully towards you in your own home.
So... speak to your husband first and agree between yourselves how you're going to tackle this, and make sure you tackle it together as a team. My recommendation would be for the two of you to take her to one side and tell her that you've heard she's criticised you in front of your children; give her an opportunity to explain her side of the story (after all, your own mother may have been misinformed). Make it clear that you and your husband are joint heads of this particular household, and as such you are both to be respected in your home by everyone who lives there. It is highly inappropriate for anyone to attempt to undermine either one of you in your own home, but especially in front of your children.
And I agree with Devlyn: your transition is about you being happy in your own skin so you can be as good a person as you possibly can be... and as good a parent to your children as you can be. It's a positive thing for your kids, because one of their parents has been struggling with a terrible medical condition that is now being treated. That can only be a good thing!
TL;DR: Tell her that she needs to treat you with respect or you'll be happy to help her find somewhere else to live.