As long as you remain married, the petition may still be approved despite the current state of your marriage because the law only requires you to prove that your marriage was entered into in good faith. With that said, you should not stay in the marriage solely for an immigration benefit... safety should be your first concern. You should consult an attorney right away to inquire as to whether you have a case under the Violence Against Women Act. This law provides relief and benefits to individuals (women and men) who have been subjected to battery or extreme cruelty by their spouse who is either a citizen or resident of the United States.
If you are in a toxic relationship, then remove yourself. Your safety and health are more important than immigration benefits. Based on the facts you've presented, you may have other options regarding your status so I'd encourage you to sit down with an attorney and fully discuss all of the facts and circumstances surrounding your family, your current relationship and your education/work history.
What was the reason for the Notice of Intent to deny? There are options if your marriage is failing. If your children were born in the US they will not be deported, although practically they may need to follow if you are. I would suggest you speak with an immigration attorney immediately.
Law Offices of Nicklaus Misiti
212 537 4407
Legal disclaimer: The statement above is general in nature, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.