What should I do about dealing with an interest-based bank to which I am indebted, while I can transfer my account to an Islamic bank that would buy the debt for a lower value? Should I continue making payments to the interest-based bank until a certain amount remains, then transfer my account? And what should I do with the surplus that accrues in my account during this period?

It is not permissible for an Islamic bank to purchase your debt by giving you cash upfront and then collecting it from you later, due to the ensuing riba al-nasi’a (interest due to delay), and possibly riba al-fadl (interest due to excess). However, if the Islamic bank pays off your debt to the usurious bank and then reclaims this amount from you without any increase, then there is no harm in that. You must transfer your account from the usurious bank to an Islamic bank as soon as you are able, and pay off the debt to the usurious bank if you cannot transfer it. You must also withdraw your money from the usurious bank and not leave it there, even in a current account, because doing so aids the bank in its prohibited activities.