You can't. You will always be the bitch, the bad-mouthing mother who doesn't take care of the kids, because their hair is not cut and you didn't wipe their mouths before a visit. You will never win this fight - at least, not with them.
Within yourself, however, you need to clear this up: if you know you are giving your children the best (in love and discipline) and you know you take care of them, and you know you don't badmouth anybody, you can hear the arguments and accusations and KEEP QUIET. Because you know the truth - as do they, but it is much easier to place the blame on anybody else, than actually doing something about it. You are painted bad, because they have no other way of making themselves feel better.
As for the kids: they know what they see and feel. You don't need to prove anything to them. Trust me, they now who left and who takes care of them. Kids aren't stupid.
BUT: you must NEVER EVER EVER bad mouth the other parent and his or her new partners. As far as the kids are concerned, their dad is the best in the world, and he loves them very much, he is just going through a tough time. (Which lasts very long, sometimes). You are not negating their feelings, you are reinforcing a love they very much need. If you bad mouth (even if what you say is true) you stoop the other parent's level. And the kids will see you see in the same light.
The sooner you accept your fate as being a bitch, the more fun it gets!! Relax, you do your best, let them say what they want to.