Should I pay off the debts of my addicted father who is unable to work, debts that exceed my financial capacity, or is the priority raising my children, building my apartment, and helping my mother?

The son is not obligated to pay off his deceased father's debts, even if he is capable of doing so. You are required to spend on yourself, your wife, and your children in a reasonable manner. If there is surplus money and your parents or siblings are in need of it, then you must spend on them. The rights of parents are immense, and honoring them is among the most obligatory duties. Part of honoring your father is enjoining upon him what is good and forbidding what is evil with politeness and gentleness, and striving to reform him and help him repent and become upright.