Neither of you can file as Head of Household if you lived together during the last 6 months of the year. So when you say that you lived together "almost half of last year" it's important to know which half and exactly how long you lived together, and whether the children lived with you or with him for more than half the year. And two people can't file as Head of Household for the same household.
You can each claim a dependent and file as Married Filing Separately if you agree to that, but why don't you file a joint return for 2016? It's almost always better than filing separately.
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