Each state is different and my divorce was a breeze compared to what you are experiencing.
Having said that I was told that it is a given that I claim the children as deductions since they live with me. No judge would sign off on anything different. I was told this because ex and his attorney were acting like this was a gift.
If you are paying the mortgage I would claim that deduction. If you are not paying the mortgage - I would leave that one go. If it were me I'd file married filing separately. I would claim daycare deductions if you are paying daycare.
Do you have a tax guy to ask?