What is the ruling on obtaining financing to pay off bank debts by purchasing a car from an institution and then selling it? Is it permissible to borrow from this money if it is not [deemed] permissible? And must this money be purified?

If the institution sells the car in a genuine sale, giving the buyer full freedom to dispose of it, then there is no harm in that, provided that the institution does not sell the car to itself. It is preferable for the buyer to sell it himself, but since the sale has been concluded and the price received, there is no harm upon him and he is not required to purify the amount. It is permissible to borrow from it even if the transaction was forbidden, because the prohibition relates to his liability, not to the money itself.