Surely the issues here are separate though?
There certainly are women who use their children as weapons against their ex’s, and claiming that a woman would only do this for valid reasons is both inaccurate and playing into the hands of women who believe they should have the right to use their children like this. It’s not palatable but it’s more common than people acknowledge here.
With this particular ex we don’t actually know whether she has said these things or whether the partner has told the OP that’s what she’s said, so it’s not possible to judge her as a vindictive ex on that basis.
The other issue though is the OP’s relationship with this man. They have taken their children out on days out, they believe somehow that the children don’t know they’re together. Of course they know. Children aren’t stupid, and they will all have friends who have step parents etc. My DS was ten when I first introduced him to my DP. I did introduce him as a friend to start with but within a couple of days he asked me if we would get married one day. 
Children aren’t stupid.
Also, telling children that they should never keep secrets from their parents is a fundamental basis for parenting. No child should ever be told that they should keep something secret from mummy or daddy. Ever. Because while it might be daddy’s new girlfriend now, it could be that they’re being abused tomorrow and how are they supposed to distinguish between those two when their parent has given them permission, requested even that they keep a secret from someone?
But secrecy is the cornerstone of this relationship isn’t it? The OP is a secret, the relationship is a secret. The days out are a secret, the whole thing is built on lies and dishonesty with an undertone of mistrust because everyone has to be secret from everyone else.
It’s not a healthy basis for either a relationship or to bring up children. And if nothing else the OP should be considering what all of this will be doing to her own child’s view of the world.