Is what I did—fasting and praying during the days I saw blood after the doctor’s examination—correct, or am I sinful?

If the total of menstrual days and the subsequent period of purity is fifteen days, and then blood appears, this is irregular bleeding (istihadah) and not a new menstruation. This is because it cannot be a new menstruation due to the absence of a valid period of purity (fifteen days) between it and the first menstruation. Nor can it be considered a continuation of the first menstruation, as the total of the first menstruation and the subsequent purity has already completed fifteen days. Therefore, what the questioner did by performing ablution for each prayer is correct, and her prayers and fasting are valid. She is not required to perform ghusl (ritual bath) when this bleeding stops because it is not menstrual blood.