Maintaining your J-1 status is more important than ever. USCIS implemented new policies that will make it even more difficult to fix an immigration status violation.
A violation of your immigration status will result in the termination of your SEVIS record (DS-2019). If that happens, according to the U.S. government, you would not be in legal status ("out of status"). If you stay in the U.S. beyond any grace period you may be eligible for, you would not be in legal status.
If either of these things happen, it can lead to the accrual of days of unlawful presence and deportation proceedings. Also, accruing days of unlawful presence can result in being barred from the U.S. for many years, or even permanently.