It is critical not to overstay your J-1 visa expiration date, even by one day. Remaining in the U.S. unlawfully has severe consequences.
If you stay past your J-1 end date without an approved extension, you will be out of status and can face:
Removal proceedings and deportation
Expedited removal without seeing a judge
Bars from re-entering the U.S. for 3 or 10 years
Possible criminal charges and jail time
U.S. Immigration and Customs Enforcement strictly enforces overstays. There are no exceptions for visa holders who get married during their program.
The only option is to depart on time. Your spouse can petition for you, but you will likely have to process outside the U.S. first.