How can an unmarried woman know that she has become pure from menstruation, especially with the continued discharge of yellow or white mucus after the eight days of bleeding, without being able to insert anything to check for dryness?

As long as you see a yellowish discharge before achieving purity (tahara), you have not yet become pure. This is because yellowish and brownish discharges connected to blood are considered to be blood, based on the saying of Aisha (may Allah be pleased with her): "Do not hasten until you see the white discharge (al-Qassah al-Bayda)."

Therefore, it is obligatory for you to refrain from prayer and other acts forbidden to a menstruating woman as long as you see blood or the yellowish discharge connected to it, provided that the total number of days (of blood and discharge combined) does not exceed fifteen days.

To confirm the cessation of menstruation, one should gently touch the area with a cotton swab.