First things first. None of us are attorneys. You must be careful of separating the advise you receive that is often based on social policy rather than legal statues. Uprooting the children again away from the other parent, home, community, school, church, etc is harmful to the children and maybe shown to be illegal even if you were born and raised in this country.
Do not remove the children from the rightful custodial parent (father) without the approval of a state court authorized to make custody decisions in the jurisdiction and venue where the children reside. At the present time and considering the lenght of time the children have been in the father's care the father is the what is considered the De-Facto custodial parent.
Here are the facts as presented :
1) the mother willingly and voluntarily relinquished the children to their their proper care giving nuturing father for their own protection. At very least, the children were at risk of foster care as well as being removed from their country.
2) the mother has no visible means of supporting herself or her children
including life's basic requirements inclusive of food, shelter, education, medical care or other training AND the children would be dependent on complete strangers for their well-being.
3) the above is just for starters .... other considerations exist including probable civil/criminal charges of child neglect/abandonment. Just because the Feds let you fly from deportation does not mean the state will be so kind on other issues. It might be helpful to brouse your FL county sherriff's "who's in jail" website to determine the type of charges you might be facing.
I'm thinking you are about 100 miles from your oldest children. Considering your plight, if I were you, my best effort would be to FOCUS ENTIRELY ON THE CHILDREN and do everything to work out all matters with the other parent for the present day and the future while reconecting with the children. Children NEED and DESERVE BOTH PARENTS.
Best to ya.