Your priority is to retain an experienced lawyer and prepare for a large cost. If you don't, prepare for a backbreaking and likely recurring cost.
I think you are going to have problems ringfencing your inheritence after ten years of marriage, funding or helping fund lifestyle, and sharing (in the eyes of court) childcare. Your other assets ie pension in danger as well (or the US equivalent).
Interest in your children and custody will increase dramatically after you leave, bearing no resemblance to his previous (and genuine) attitude to childcare. The court will not care at all on last attitudes, barring evidenced abuse or neglect, so when he starts playing doting parent, you need to be prepared for that and deal with what you have now instead of arguing with judge that he didn't do any cleaning, never picked up kids et
-Make every conversation you have with court child focused and not relationship focused.
Think of one positive thing you can say in court about him.
What you described will not meet threshold for concern over children from either side, however, his leaving your children in the care of someone unknown to you and not vetted could backfire in you as well as him, so tell lawyer and do what they say.
If you have ever had concerns over his child care this is the point to discuss them with your lawyer.
He may claim you been funding and supplementing his lifestyle and living needs. Your lawyer needs to protect you on this so you don't end paying to him lion's share and he gets FT of your kids. Believe me, he will most likely go full custody and will get a shared arrangement at least based on what you've said.
Get your ducks in a row before leaving: documents, any evidence.
As a woman, there is a sword over your head that not hanging over his, if you do go to court: