I agree with Attorney Dzhamgarova. If you choose to leave on your own, keep track of your receipts as proof of the date you left the US in case you need to prove you left at a certain date later. Good luck.
Legal disclaimer: The statement above is provided by CC Abbott is based on general assistance and not intended to be a legal opinion because not all the facts are provided. The person requesting information and all others reading the answer should retain an attorney who is permitted by the state bar within the jurisdiction who can examine the complete facts and provide a legal opinion on your case. All information provided in the above answer and other information provided by CC Abbott does not create an attorney/client relationship within any state of Federal law.
You should not be intimidated by ICE officials. Be firm, request a court hearing if you are eligible for it, and do not sign anything before you consulted an with an immigration attorney.
If you are granted the privelege of voluntary departure (without safegards) by the immigration judge, you leave how ever you want. There is no interaction with ICE, other than verifying the departure at a consulate or at the border.