Really the date that matters is when your program ended. You need to file before then if you want to change status.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Yes, it is possible, but you need to take immediate action. The controlling date is the expiration of your DS-2019 (your program end date), not your visa. A J-1 alien can remain lawfully in the United States for 30 days past his or her program expiration date. Thus, any application for a change of status can must be filed within that 30 day time period.
Consult with an experienced immigration attorney for case specific advice.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Call +1-561-478-5353 to schedule a consultation with Mr. Devore.