I can't tell you anything about the filing deadlines for forming an LLC in New Jersey, but I am one of those CPA's that can be tough to talk to during the day. However, if you catch us sitting on the couch late at night on bogleheads.com you might be in luck.
File form 2553 along with your first tax return, which will be an 1120S and follow the instructions below which are from the Instructions for Form 2553:
When filing Form 2553 for a late S corporation election,
the corporation (entity) must enter in the top margin of the
first page of Form 2553 “FILED PURSUANT TO REV. PROC.
2013-30.” Also, if the late election is made by attaching Form
2553 to Form 1120S, the corporation (entity) must enter in
the top margin of the first page of Form 1120S “INCLUDES
LATE ELECTION(S) FILED PURSUANT TO REV. PROC.
The election can be filed with the current Form 1120S if all
earlier Forms 1120S have been filed. The election can be
attached to the first Form 1120S for the year including the
effective date if filed simultaneously with any other delinquent
Forms 1120S. Form 2553 can also be filed separately.
The IRS has been very flexible when these instructions are followed. If you act like an S Corp (i.e. file an 1120S return) it seems almost a foregone conclusion that your election will be approved. I'll be shocked if the IRS doesn't accept the election. Not sure how you'll fare with NJ if they truly won't accept LLC formations after March 15.
A note about SE Taxes. An S Corp may
provide an opportunity to save on SE taxes, but the IRS' main focus will be on "reasonable compensation". You will want to familiarize yourself with the concept (if you haven't already) to make sure you won't end up in an audit on this issue. The most famous case was from an S Corp owner who paid himself $24k and took distributions of $200k. He got audited, but being "brave", told the Tax Court Judge that "no one can tell me how much I have to pay myself". The Judge disagreed and determined that his compensation was to be around $125k which resulted in some serious penalties for not paying his payroll taxes timely. The kicker is that the S Corp owner was a CPA. Find a good CPA who is familiar with S Corp taxation so that you can avoid the problems of not getting things right the first time.