Is the fast valid, or is purification required again after seeing dryness, then the cotton swab becoming stained, then a drop of blood coming down?

You mentioned that you checked the area after bathing from menstruation and found a trace. If it was blood, then it is menstruation, which invalidates the fast and necessitates ghusl. If it was a brownish or yellowish discharge, then it is not considered. You must perform ghusl after the last blood flow ceases and make up for the fast of the day on which the drop of blood appeared.