Is it permissible to continue trading in an account that has converted to an Islamic account after overnight interest (فوائد تبييت) was deducted from it, or is it obligatory to give away in charity the previous profits that did not include overnight interest?

There is no objection to benefiting from money earned through lawful means, free from usury, and there is no obligation to donate it as charity. If the money was acquired through unlawful means without prior knowledge that the account was non-Islamic, there is also no objection to benefiting from it. It is advised to avoid dealing with usurious institutions unless there is a necessity and no Islamic alternative is available.