It's essentially like putting the property on layaway only you get to at least take possession while you pay the contract.

She offered to pay June's rent out of her own pocket without taking possession of the property or signing a lease. 

There is no paperwork we use upon early termination; when we receive the keys to the unit from the tenant we have legal possession.  

If I have a tax deed, and I have removed any questions about possession, either by having the taxpayer sell me his remaining interest, or by a successful ejectment, what do I then have in terms of ownership?  

They signed it when the lease was renewed and I referred them back to it often and even made sure they had a fresh copy in their possession a few months prior to lease termination.Just to provide some context for discussion, they had small kids and dogs and I expected some amount of wear and tear to the floors.

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. 

Do they care and respect other people's possessions?

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.

You can also warn them that the water may damage their possessions and you are not responsible for what happens.