How to win primary placement in 'he said/she said' case?

I live in New York state My emotionally abusive husband threatened to take our 20-month-old twins and not return with them. He also called CPS and filed false neglect allegations against me to try to get them taken away from me. Nothing came of that.

I left the matrimonial home on 10/22 with the babies and went to a safe house for abuse victims. 

He allowed me to become illegal in this country by refusing to file the paperwork for me to get a Green Card after nearly two years of marriage. He was hoping I would be deported and he would get the babies. He also took the babies passports and all their documents.

I have petitioned for a Green Card under VAWA and they have said I have established a prima facie case.  

I have no witnesses to his verbal/emotional abuse or his lack of parental involvement. I have always been the primary caregiver. I am a stay-at-home-mom. He did nothing besides change the occasional diaper and feed them every once in a while. But I have no proof and he told the judge the opposite. 

How can I establish that I was the primary caregiver and that he was abusive and an uninvolved parent? How can I win primary placement when he is willing to lie under oath and has witnesses willing to do the same for him? 

The issue is, you cannot expect to win it all since you have no proof. Emotional abuse is very hard to prove and just because you have a fight does not mean its emotional abuse. Are you planning to move back to your home country with the kids? If you were a stay at home mom and he provided for you, and the kids, that does say something. Most of the time SAHM's are the more involved parent that does not necessarily mean Dad is unfit.

You may have to settle for joint custody however you should still get child support and day care allowance so you can enter the work force if you do get a green card.  Although you may not be totally happy with not getting sole, sole still gives Dads some rights anyways. In this day in age it is very hard for a married mother to walk away with sole custody. Dad will still likely get unsupervised visits with the kids regardless.

He would shout at me for hours while following me around the house, even into the bathroom, calling me the worst possible names. He would grab the babies from me and hold them while shouting at me. It wasn't any ordinary fighting and I kept silent while he did this. 

I would like to take the kids and move back home but he has taken their passports to prevent this happening. Can I petition the court to allow me to do this? If I did he would object strongly because he told the judge that's why he took the passports. He put me in a position where I would be forced to leave this country, even though we were married almost two years, and took the passports so I could not take the babies with me. That's not abusive behavior? 

He bought the babies necessities, but I had to get money from my parents to get personal care items for myself because he refused to give me any, ever. He claimed the things I wanted (like vitamins) were not necessary. He has just sent back the toddler vitamins I left at the house, to let me know he will not be giving it to the babies (because he grew up without it so it's not necessary).

I am not trying to take away his right to see the babies. I just want them to have a stable home base with me and not be away from me for a week at a time. 

There is probably no way an American court is going to allow you to relocate out of the country with the kids. The kids are American citizens. Seems like he has the passports for a good reason. You need to be realistic here. The kids were born here, and Dad and his extended family are probably here. The courts are not going to remove the kids from their current environment. I suggest you see an immigration attorney to see if there is a way to remain in the county legally. If you move the kids away to your own country surely Dad will likely not see them again. I am not sure if even being married for less than 2 years would get you legal citizenship anyways.

It is he said she said here. You have no proof of physical abuse. He can also make outrageous claims but without any proof a judge is not likely to believe him either.

I cannot imagine a situation, short of evidence of attempted murder, that would cause a judge to allow you to remove the children to another country at this age. They are US citizens and thier father has the right to a relationship with them.

You MIGHT be able to get supervised visitation if you can prove he is a danger to the children. But that does not seem to be the case.

You need an immigration attorney; your divorce and custody situation is a cake walk compared to your immigration status. If you've been here illegally for all this time. You may be banned from entering the U.S. for 10 years.

VAWA is probably your best option; it may be your only option. 

According to her post she is in NY