I think being 23 and just having a whiff of love for the first time, I'm probably not qualified to help. But I'll give it a shot if it can give you any perspective.
You guys seem to have been through everything together. The abuse, the fights, having kids, everything. While the romance and the love may have died for you a while ago, the history you have and the bond you have made doesn't make letting go any easier. And he will always be in your life, because of your kids, so it is not like he will just up and disappear from you because you have to keep some form of contact with him. It doesn't sound like a very easy situation and it's another reason why I'm glad I'm holding off on marrying and having a family for when I'm out of school (28-29 years old at the minimum).
Being freshly apart and with new partners already, it's almost like it's a game to see who is winning the break up. He's asking you how you two are doing, fishing around, being annoying no matter how happy he seems with the other. He feels the same way and I'm sure it hurts him too to see you with somebody else. I wouldn't respond to those kind of pokes and prods, I would only discuss things with him that have anything to do with your kids as I'm assuming that you and your ex are sharing the responsibilities of time spent and care. Your private life is really none of his business and it only is if you are freely offering up the information.
For me, moving on has always been accepting the fact that you do not need them in your life to be happy. You can still love them and appreciate them for who they are, but know that they aren't in control of your happiness and the only one that can make you happy is a pretty important concept. Obvoiusly you are going to hate him for the things that led to your break up and maybe not love and appreciate them, but you don't need to smear him and just think of things that made you angry to help move on. Accepting that you both did things wrong (it does take two for a relationship to crack, you had a hand in it too), it didn't work out, and you don't have the same feeling for him anymore is fine. Nothing to be scared of. And let's face it, your ex is going to be better in certain respects than your current partner, vice versa, and it's natural to compare the two. Concentrate on your current's positive aspects (although don't turn a blind eye to any bad things) and focus on enjoying the relationship and the now (when you aren't too busy working or taking care of your children). This is important for you and your partner, because looking back on your ex and everything can have a negative impact on what you have going on currently.
What's done is done, nothing can change what has happened. Learning from what went wrong with your last relationship and applying those lessons to the current one can only enhance your experience and bring you closer to that happiness you desire. I guess I'm talking up your current partner alot assuming he isn't a low level choice and rebound, especially at your age and considering you have kids. If you have a good thing going, give that your all. It's not going to be easy and you are going to think of your ex time and time again. Taking it one day at a time, not giving in to his fishing and nosiness, and spending the time you have focused on bettering yourself, your kids life, and your relationship is the only way you can come out gaining from your experience.
Waking up next to a beautiful girl,
Step outside and say hello to my beautiful world.