Is there a difference in the method of ghusl for women and men after intercourse, such that a woman begins by washing her body with water only, then performs wudu, then uses soap and shampoo? And is there a specific supplication that should be recited when intending purification?

There is no difference in the manner of purification between a man and a woman. Both must wash their entire bodies with pure water, and it is recommended for them to perform ablution in the manner of the Prophet, peace and blessings be upon him, as was established from the hadith of Umm Salamah, may Allah be pleased with her, and as affirmed by the Permanent Committee. The use of soap does not affect the validity of the ablution. There is no specific supplication to be recited during ablution except for saying "Bismillah" (in the name of Allah) at the beginning, and afterwards, the prescribed remembrance after ablution should be recited. As for uttering the intention, there is no need for it, as its place is the heart.