It's tricky but you sound like you have a good handle on it.
It's very early days for the kids.
Like any child, they might it hard to accept anyone who takes their mother's attention away.
Depending on the children's ages or what life stages or of grief they are going through, there could be a sort of 'my Dad just died and now my mother has a boyfriend and she's really besotted with him and thinks the sun shines out of his arse, he can do no wrong in her eyes, nobody really cares about me anymore'.
So I'd proceed with caution and take a wide perspective of where every child is 'at' in their lives (recently moved to college etc,).
If your previous partner love-bombed and controlled you then there is a shift taking place in your own dynamics.
Knight on a white horse comes to save me because I was locked in the tower.
There will also be a shift in the dynamics between your ex BF and because now it's a reality that you two can share a life.
Proceed in that relationship with caution too.
You don't want to accidentally get locked in another tower.
Take your freedom of being alone for a while.
In case it's a fear of being alone that underpins the urgency to get certainty put on this new relationship.