There is no safe way for you to travel in the US until you have corrected your immigration status by obtaining another valid visa or leaving the country. You should contact an attorney to make the best arrangement you can.
If you want to travel from Alabama to New York you may have a problem if immigration officials request proper documentation, regardless of how you travel.
I am not sure what new immigration law you are talking about, but immigration laws are mostly federal and thus the same in all states. You should discuss your immigration status with an immigration attorney so that you know all your rights and options for adjustment if possible.
You are always at risk of coming into contact with immigration agents and being placed in removal proceedings for having overstayed your visa, but in practice many immigrants travel throughout the country without ever coming into contact with an immigration agent.
We can not give you advise as to how to avoid immigration authorities, but they generally are not looking to prevent people from traveling from one state to another.
Legal disclaimer: This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Do not rely on this advice without speaking to an immigration attorney in detail about your case. This message does not create an attorney-client relationship.
Any person who is out of status risks being found every day they exist within the United States. The "immigration alw" you speak is not a law it is a comment made by USCIS that people in deportation may be reviewed. The best help I can give you is to contact an experienced attorney to review your case and determine if there are any options available to you. Be careful of notarios who will say they can "fix" your status but will only steal your money and make false promises.
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.
1) There is no new immigration law.
2) No one can guarantee that you can travel out of status without any problems.
J Charles Ferrari Eng & Nishimura 213.622.2255 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.