I went through a divorce a few years ago. Some lessons learnt from my side which may or may not be applicable to you.
Context, we were together 10 years, married around 3 and a bit.
Exceptionally good relationship and very amicable in terms of divorce, and even then, we had some moments.
I would absolutely not co-habitat. It is confusing for you guys, for the children and will make moving forward harder. My daughter was around 3 when we split, and we did a 2-2-3 day split. Which means we each got every second weekend with her. Once she starting going to proper school, we did 1 week on 1 week off.
Something I noticed for myself and other guys is that initially the divorce does not impact us that much. For the first few months we tend to feel free most of the time (there are still those hard, depressing times as well) and we try live our life. Woman I found struggled for the first few months and then got better.
The actual divorce and everything it entailed started to crush me around the 9 month to 1 year mark and stayed with me for about another year.
This means that for a new relationship, you are making it very very difficult for that to work. You will likely have an awesome "honeymoon" phase and then realize that all relationships can be hard. You will compare your new relationship to your ex wife, you will start to have regret, resentment, anger, depression and it will really take a toll on you and your new partner. Because it is an old ex, you might also be tempted to progress that relationship faster than you normally would, which means she will start attaching strong feelings sooner and when the divorce hits, it will really hurt her. This is all my opinion and based on experience and what I have seen, but it seems to be pretty likely and I would strongly advise to not jumping into anything and rather focus on figuring out the divorce, what went wrong, how to improve yourself, prioritize your kid and give everyone time to heal.
You will not get proper time if you co-habitat and it will just drag things out, ruin the amicable relationship and confuse the kids.