Sauron

[Maedhros]

Therefore, I cannot see why Sauron was forced to depart into shadow when the Ring was separated by him, and why he could not continue the fight. The text only suggests that he needed the Ring to create for himself a new being, which would keep the whole point of the Lord of the Rings valad, but it does not suggest that if the Ring was destroyed, Sauron would depart forever.

You forget about the concept of Incarnation of a Maia. A maiar is an ainur, and therefore an ëalar. (a being that is naturally discarnate) An ëalar can take physical form to interact with other beings in the physical plane as per in the Published Silmarillion. Different maiar had different levels of power and because Sauron had more power than your average maiar, he could incarnate himself in different forms, but as he did evil, his control over that ability lessened.
From the Published Silmarillion

Sauron was not of mortal flesh, and though he was robbed now of that shape in which he had wrought so great an evil, so that he could never again appear fair to the eyes of Men, yet his spirit arose out of the deep and passed as a shadow and a black wind over the sea, and came back to Middle-earth and to Mordor that was his home.

Because Sauron, like Morgoth, became dependent on their hröa, after their hröa was slained, they could remade a new one, but with each attempt they lost part of their power, but because of the fact that Sauron had put so much of his power in the Ring, he could reclothe himself because of the presence of the Ring, but each time it would take more effort and time.
For more information you can read the Osanwe Kenta essay