Great advice bares repeating.
Ask around for recommendations on a good family lawyer if possible.
Document everything. You may not need much of it, but what you do need you'll have, and the courts will see rather than just hear how seriously you take your responsibility toward your children. I've seen a father lose a daughter to a mother that stuck up for her daughter's molester years before. He lost because she had documented every minute she had with the child for visitation and was determined to be the supporting parent.
Keep your cool. Keep your cool. Keep your cool. The courts are not moved by passionate anger but by facts. Even hearsay can lose you ground and incline them against your case. My own brother was not allowed to be around my kids during my divorce because my soon to be ex-MIL told the courts confidential details of counseling sessions she had with him (spiritual counseling). He was bi-polar, manic depressive, psychophrenic, and though nothing he shared had to do with thoughts of harming anyone other than himself, rather than dig into it the judge simplified it by saying he couldn't be around them. Her goal was to circumvent my time with them because my brother lived in the same house as I was staying at the time.
Do not bring the kids into it by talking about their mother poorly. There may be things you'll have to address of course. There may be times you'll have to table the discussion until you can do so with a level head. You can talk about differences of opinion. You can talk about why some rules are different with you than with mom. These are okay. "Because mom is a psycho bitch" is not, no matter how true. My kids knew of my extreme dislike of their father and I made them aware I would always try to deal fairly despite that. I kept them involved with the facts of the process and let them draw their own conclusions and ask questions, keeping things within their grasp of understanding. And while their father and grandma fed them and the courts a steady stream of bullshit, they got facts from me. My daughter turned against me for many years. It hurt like hell, yes, but I stayed consistent. As an adult herself now, she's come back to me. She's told me she understands why I did what I did. And even moreso, she sees them for what they are, and that is the prize. Not revenge, but seeing her gain the ability to think for herself, to see people for what they are by what they do. Coming to that understanding on her own will do her much more good in her life than just believing dad is an ass because mom said so.
Bottom line, this woman is still their mom and will continue to be so, whether you're helping your kids with the transition or making it worse for all involved by making them hate her. And I'm sure she loves them though she's obviously not making them a priority right now. She will regret that later. You'll have no control over what she has to say to the kids about you. You most likely can't stop her from integrating them into her new life with her new man. You can stay on top of their safety and well being however, and sometimes that may need to be done through the courts if rational discussion doesn't work. (Also, the court really doesn't care who she's living with, as long as there is no threat posed to the kids).
I certainly wish you the best of luck. I hate seeing a good parent get screwed over. I hate even more seeing a good parent get screwed even worse because they were too lazy to take sound advice, fail to protect themselves, thinking that being in the right will carry the win. DONT BE THAT GUY! The courts don't care who did what to whom. They care about the children's welfare, and yes, they care about balance when feasible.
God, so much still popping into mind ... I better stop before there is no turning back...