Merely touching one's private parts doesn't break Wudu for Shia Muslims; it would however break one's wudu for those who refer to themselves as Ahlus Sunnah. Yes, if one masturbated causing ejaculation this would break the wudu and one would require ghusl al janabah.
May Allah grant you success
Either is fine.
May you always be successful
No need to be dry before starting the Ghusl. We can perform Ghusl on wet body right after the shower.
This is one of differences between Ghusl and Wudhu. In Wudhu we must make the area of wiping (over the head and the feet) dry before wiping over it, but Ghusl does not need any part of our body to be dry.
After the Najasat is removed, shower water falling on the Najis areas is enough immediately to make your body Taahir with out need to go out of the shower and re enter.
Washing your body with out intention of Spritual washing which is called Ghusl, does not mean that you will not need Wudhu for Prayers. Wudhu is must before performing Prayers. If you perform obligatory Ghusl like Ghusl of Janabah, then you pray with that Ghusl with out Wudhu. If you perform a recommended Ghusl like Ghusl of Jum'a, then you can pray with it with out a need for Wadhu.
Just shower or washing the body does not mean valid Ghusl.
After you perform Ghusl or Wudhu, your state of purity ( Taharah) continues unless it is broken by wind, urine, stool, sleeping, etc. Any breaking of wind will invalidate the Wudhu and Taharah.
You can perform Wudhu just before starting your shower. It can be also done after shower but after you dry your head and feet where wiping on them is compulsory.
Wudu is a worship and cannot be replaced by ordinary washing or shower. The recommended ghusls are described in books of fiqh and you cannot make ghusl out of those described.