Is purification by dryness valid if the period has been prolonged and the white discharge has not appeared, knowing that waiting for the white discharge may lead to missing prayers?

Purity is recognized by one of two signs: the cessation of blood and dryness of the area, or the discharge of the white fluid (al-qassah al-bayda'). What you mentioned about relying on dryness and not waiting for the white fluid is correct. So, if menstruation begins and then ceases, and the area becomes completely dry, the woman has attained purity, and fasting and prayer become obligatory for her. She is not required to wait for the white fluid after dryness. However, if the area is not dry, or if there is any yellowish or brownish discharge, the woman should not rush until complete purity is achieved or the white fluid is discharged.