There isn't an "exit tax"; but the prepayment of estimated taxes due is only required on exiting residents. It therefore earned the nickname, exit tax. Whether you leave or not does not determine if a tax is due. It can impact when and how it is paid.
The following are state taxes in NJ that are assessed upon sale and transfer of title. There are exceptions and exemptions.
Realty Transfer Fees apply to all sellers. This has nothing to do with leaving the state. Many states have a similar tax by similar or other names. This is paid by the seller.. The tax is determined on a scale, using a formula. (Estimate about $5700 on a $650,000 sale. Lower if the seller is aged 62+.)
A Luxury Tax (AKA - mansion tax) is assessed on buyer for all properties at or over 1M. It is 1% and paid by the buyer at closing. There are times when buyers make an offer on a property and ask the seller to pay this tax on their behalf.
The "Exit Tax" is not a special or additional tax. At closing, for exiting NJ state residents, an estimate is made for the amount, if any, of income tax that might be owed based upon the sale and transfer of the title for that property. This would include, but not be limited to capital gains. If there is an estimated tax due, a declaration and payment must be made at closing. Residents are allowed to file exemption paperwork. This isn't an "extra tax", it is NJ's way of assuring taxes due will be collected, since any collection activity outside the state would be more difficult.
I am not a tax planner or advisor. Consult a tax advisor for advice for your specific situation and whether or not a prepayment of estimated gross income taxes might be required.