How can a father be convinced to pay off his many accumulated debts before his death, despite his lack of understanding of the seriousness of the situation, and how should one deal with siblings who do not cooperate in solving the problem? And should the father be given his full salary to pay off the debts?

You have done well by caring about settling your father's debts. Islamic law has paid great attention to the matter of debt because its owner may claim it from the debtor's good deeds on the Day of Resurrection. Moreover, whoever borrows with the intention of not repaying will meet Allah as a thief. The Prophet, peace and blessings be upon him, refrained from performing the funeral prayer for anyone who died owing a debt, which indicates the seriousness of the matter of debt, and that procrastination in repaying a debt while being able to do so is forbidden. Discharging the rights of people is done by fulfilling them to their owners, or to their heirs if they have passed away, or to the public treasury if they have no heirs. If the creditor dies, the right passes to their heirs. Your eagerness to settle your father's debts is a righteous deed, even if it is not obligatory for you. If you were to ask the creditors for forgiveness, it is likely that they would agree, in consideration of your father's circumstances.