Is the described method for ghusl correct, focusing on washing the hands three times, then the private parts and between the buttocks, then washing the hands with soap, then performing wudu, then pouring water over the head three times, while focusing on under the armpits and the navel?

As long as you make the water reach your entire body, your ghusl is valid. It is a ghusl of sufficiency that fulfills the obligation, and it is performed by making water reach the entire body along with the intention. There is also a complete, recommended ghusl in which the person performing ghusl follows the manner of the Prophet's ghusl, peace and blessings be upon him. You must strive against whispers (waswas) and not give in to them.