What is the way to pay off a debt I am unaware of and whose details I do not remember, and is giving charity with the intention of paying it off considered sufficient, given my severe panic and constant thoughts of the torment of the grave?

The fear of the torment of the grave is a good thing, and it should motivate a Muslim to fulfill their obligations and perform their duties. As for debts, the basic principle is the acquittal of one's liability unless one is certain that it is encumbered by a debt, in which case it is obligatory to repay it. If you know the owner of the right, it is obligatory to render it to him. If you do not know him, or are unable to deliver it to him, then give it as charity to the poor and needy. Izz al-Din ibn Abd al-Salam said: "The basic principle is the acquittal of one's liability, for Allah created all His servants with their liabilities and bodies free from His rights and the rights of His servants until the causes for their obligation are established." Giving charity out of fear of an overlooked right is a good deed and one will be rewarded for it.