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.