I don't know the answer to your personal questions but to legally work you need a valid work authorization. I would speak to members of the shelter about where people typically go after there allowed time there is up.
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.
I would contact a public interest/human rights/non-for profit organization in your area to help you with your case and discuss options with you. U visa cases is something they typically handle.
This is not legal advice and a client attorney relationship is not created. For a free consultation call (718)234-5588.