What is a Demand for Possession?

An eviction is a suit for possession of the property.

If tenant has left ad turned off utilities and most of his things are gone, you can claim the unit as abandoned and take back possession. you may be required to store his belonging for a period of time. your last option is to convince him to sign over possession of the unit, which may or may not require you to offer to pay for his moving expenses or cash for keys.

When we close and take possession we will be requiring they sign a one year lease if they would like to stay (they've lived there for 12 years so I am not too worried that they'll put up a huge fuss).

Which means that should anything happen to it, 1) you already made a deposit that is not going to be returned, and 2) you have no control over the property for at least a month (of not two, if not even more if it is occupied and you have to file a writ of possession once you make the full payment and receive the deed). 

He entered a judgement for possession in 10 days.

Is there a tried and true method to ensure that the current clients know who to send their rent to, after the the new owner of a MHP takes possession?

New Jersey Statutes give a right to a Judge within his or her discretion after Judgment of Possession has been entered and after the Judge has made a determination that no further Order to Show Causes will be issued to give a tenant time to vacate from a tenancy.

This will give you possession and time to find the necessary financing without feeling a great deal of pressure. 

If she moves out of state, then you can take possession of the apartment through abandonment.