What must I do after having broken an oath sworn upon the Quran that I would not return to a sin, considering that I do not recall whether I said "Inshallah" (God willing) at the time of the oath, and is it permissible to feed the needy in installments?

The cure for sin is repentance, remorse, abandoning it, and resolving not to return to it. The mere decision to swear by the Qur'an is not an oath unless it is actually uttered. If you swore an oath and broke it, and you doubt the existence of an exception, then expiation is not lifted unless the exception is your habit. If making an exception is your habit and you doubt whether you made one, then there is no expiation upon you. What you receive in Eid gifts, presents, and pocket money from your parents is your property; you are permitted to pay expiation from it. You are not permitted to fast if you have what can feed ten needy people.