Estate Law Questions? Ask an Estate Lawyer.
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You cannot get the house unless the elderly lady gave it to you by gift or will, or if nursing home settles with you to have some debt paid and house released, and her estate is not claiming the house. You need to work with the cousin on this issue. Normally as a non relative you cannot probate unless no one else seeks to do it and no one objects within the family. You may want to find out if the house has equity and size of debts and mortgage, since the debts may exceed the equity.
You do not have rights to the house from the facts you mentioned unless the heirs decide it to give it to you.
Lender and creditors would have priority over her direct heirs who have priority over you. Your friendship/occupancy do not translate into rights in the property.
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I realize its not what you want to hear but adverse possession would not apply here, and there is no title vested in you.
adverse possession is a rare theory which applies to occupancy for 10 years without consent in property