I agree with everything below. Especially where your daughter is having behavior problems, she is clearly still in the acute trauma phase of processing your divorce and her new life. 110% of your time with your kids needs to be focused on them and they don't need anyone new in their lives, especially someone who they will see as competition for daddy. Keep your love life separate for a long, long time. Like years. Chances are, just based on the timing of the situation, your current guy could very well likely be a short-term relationship (Mr. Rebound) so there's no need to introduce them to someone who might not be there long-term. And if this relationship lasts, there will be an appropriate time later to integrate him into your family as you're planning on a permanent relationship.
Also make sure that you don't let the new relationship occupy too many of your thoughts while you are not on kid-free nights or weekends. It's easier said than done, I know! I was a single mom and even though I tried to not let my relationship with my boyfriend encroach on my everyday life (and I had no kid-free time) there were definitely times when I was trying to play cars with my son or toss a ball with him when I was distracted by thoughts about my bf, and that wasn't fair to my son at all. My situation was a little different because my son never met his birth father, and my bf and I had been friends for a year through our kids (his daughter is the same age as my son, it was just kiddie playdates at first) but even with all of that groundwork, my son still flipped out when we started dating. He was 3 and would say things like "don't touch my mom" or "oh why his HE here AGAIN." Unfortunately, they're still not close even though we've been married for 7 years and have known each other for 10.
So...take it very, very slowly, for your sake as well as your children's. Enjoy the company of your guy, but keep it as something you do on your own time and don't try to integrate him into your life with your kids.