You really do not have to worry!
First of all, at this point his threats are just words. Until he actually files something, you will do best for yourself if you just ignore him.
If he does actually file an action requesting a modification of custody, he will have to show the court that there has been some change in circumstances that makes the modification necessary. Increased child support is NOT a justification for modifying custody. Based on what you've stated, he's not too likely to be able to address that first hurtle. But even if he does, he would still have to be able to prove that the arrangement he is seeking is in the children's best interest. The point is, just because he asks for it doesn't mean it will happen.
Again, until he files something, just ignore the threats. If he actually does file something, just hire an attorney to assist you. But for now... stop worrying.
Good luck living your life as usual,
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
Ignore what he says. At this point you have NO concerns and no legal problems. You will worry yourself to death asking for help for a problem that does not exist.
When and if he files something, you need legal help. Someone saying they might do something happens every day, usually when people have no plans to do it.
You just won one battle with him. There is no cause for panic.
He may just be trying to intimidate you, but even if he files an action for custody, based on the facts you've provided, it is unlikely that he would prevail. None of us have a crystal ball, but if you have always been the children's primary custodian and if you have been the parent who has nursed the children when they were sick, made sure they bathed, gotten them ready for school, attended parent teacher conferences, driven them to and from extracurricular activities, attended school plays, basketball games, dance recitals, etc., and if he has maintained little contact with the children, does not have any means of keeping heat in the home, and moves around from place to place, it does not sound as if he has a very strong case (to say the least) to seek custody of your children - nor does it sound like he can afford an attorney. The facts, based on what you've shared, are in your favor.
All that said, it would be wise for you to consult with an attorney - not just about the father filing a modification of custody action, but about visitation parameters being put in place that would protect your children, so that they are not left alone in the father's home, so that they are not in a home without heat in the winter, and so that they do not drive with the father until such time as he has addressed whatever medical issues (i.e., a sleep disorder) he has that may be causing him to fall asleep at the wheel, if that is happening. Best of luck to you.
This answer is offered for informational purposes only. It is not offered as nor does it constitute legal advice. This answer does not constitute an attorney-client relationship. Do not rely on this answer in prosecuting or defending against any criminal or civil legal action. Speak to an attorney in your area about how to protect yourself and your interests.
Simply because the father of your children is a police officer does not give him any more right to a modification of custody than any other person. He would first need to show the court that a change of circumstances has occurred warranting a change/modification of the custody order AND show that the change is in the Best Interest of the Children. Based on information in your question there is apparently significant evidence that a change/modification of custody should not occur.
You should make every effort to document the threats and poor parenting he has exhibited for use IF he actually files an action to change or modify custody. This brings me to my final point -- until he ACTUALLY files a legal action there is nothing for you to worry about. At this point there is only the threat of legal action. If he files an action later, you have had this time to get your defense in order. I hope this answer has been helpful.
The information provided in this response to a question is not legal advise and is provided only for general information purposes. My response should not be taken as legal advise as no attorney / client representation exists. Additionally, the information given in this answer is specific to the State of Georgia only and should not be applied to any other state.