Is my ghusl valid? Please help!

The fard acts of ghusl is washing nose, washing mouth, and water in all parts of body. I just finished my menses (period) so I had intention to make ghusl because I want to pray and start fasting (it is holy month of Ramadan now). I went in the bathtub, washed my private parts with water and soap. Then washed my hands with water and soap. Then washed in my nose 3 times and mouth 3 times. I then put water on my hair with shampoo and washed it very well. Then put soap all over my body (except my private parts because I already washed them) and washed with water. I got out of the bathtub and wiped my body with a towel, put clothes on, and hair dried my hair using hairdryer. Then I did wudu and put face moisturizer on (I have to put it on because of my dry skin). Is my ghusl valid correct? Please tell me if it is! I want to pray, start fasting, and read Quran from mus'haf. Thank you.