From reading your post, it's clear that two key factors are crucial for your mental health: acknowledging you didn't end the relationship and recognizing she's already moved on. Her swift transition to dating someone else indicates a prior disconnection from your relationship.
Given your narrative suggests it was her choice to end the relationship, it's logical to conclude that she should be the one experiencing trauma based emotions, not you.
Healing from a bad breakup takes time, and there's no shortcut to recovery. The best approach is to seek professional support and surround yourself with healthy friendships, ensuring you're available for your kids on all levels.
Expect the emotional turmoil to intensify before it subsides, as it's still a fresh and new situation. It's going to be a rollercoaster for a while, but one day, it won't be.
Remember, it's okay to ourn the loss of the relationship and your envisioned life. However, it's equally important to recognize that if she didn't want to be part of the relationship, you shouldn't spend your life broken hearted over her.
Focus on being a good Dad and prioritize your own well being. This will enable you to navigate the challenges ahead and emerge stronger. Best of luck.