What is the jurisprudence of priorities in settling debts for family versus marriage? Did the Prophet, peace and blessings be upon him, advise someone complaining of poverty to marry? And is debt considered poverty, such that my situation applies to it?

You are not obliged to pay your family's debt, but it is commendable to do so. You must hasten to marry if you fear falling into what is forbidden, because purifying oneself and protecting oneself from the unlawful is an obligation, and marriage is its path. There is no authentic hadith (definition) stating that marriage brings wealth, even though some narrations from certain Companions have been reported to that effect. Not every debt is considered poverty; if the debtor is able to pay, they are not considered poor. As for someone who has a debt they cannot repay, they are poor, and it is permissible for them to receive from charities. In your case, the debt is not considered poverty because you are not the original debtor.