The children can be claimed by either of you or split between both of you. Only the child that is a biological relation to the taxpayer claiming them on their return can have HOH filing status.
Since you both have income and both are able to claim a biological child on their tax return, you need to do two tax returns to see which one gives the best outcome. Only one of you can have Head of Household filing status since you are both living in the same household.
