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. 

Once she has possession of the property and title, she can sell it, lease it whatever.

My lease has a clause that says if they do not take possession of the premise in seven days (a.k.a move in inspection with me) then they loose deposit.

I would have your realtor have the seller give the tenants your name and phone number prior to you taking possession.

Do they care and respect other people's possessions?

In California, you have to add "and all in possession" wording to all eviction paperwork to cover those who is not on the lease document but who can claim tenant rights by the length of their stay.