Is it necessary to fast after the cessation of blood, or after seeing the white discharge, in the case of an irregular menstrual cycle, as happened in the months of Rajab and Sha'ban?

The dryness of the area, indicated by inserting a cloth and it coming out dry, is a sign of purity from menstrual bleeding. If you have confirmed the dryness of the area, then it is obligatory for you to perform ghusl (ritual purification bath). The brown discharge you noticed after purity is a kudra (turbidity) and is not considered menstruation by Islamic law. It does not necessitate ghusl, nor does it affect the validity of fasting in Ramadan, but it does nullify ablution (wudu').