What is the ruling on the prayer of one who has perfumed themselves with a fragrance or cosmetic product whose contents are unknown and which may contain volatile alcohol that does not remain on clothes?

Every perfume product lists its ingredients on its packaging, including the percentage of alcohol, if any. Otherwise, it states that it is alcohol-free. If it does not state this and we do not know if it contains alcohol, there is no harm in using it, because the default ruling for things is permissibility until a sound evidence proves otherwise. As for the drying of a garment from perfume containing alcohol, this does not purify it; rather, it must be washed with water.