In the books he isn't a giant eye but has a spirit form and a physical form, even in the Third Age.
But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places; and he took no visible shape again for many long years.
The Silmarillion, Of the Rings of Power and the Third Age
'Yes, He has only four on the Black Hand, but they are enough,' said Gollum shuddering.
The Lord of the Rings, The Two Towers - Book 4 - Chapter 3 - The Black Gate is Closed
...the year 1000 of the Third Age, when the shadow of Sauron began first to grow again to new shape.
J.R.R. Tolkien, Letter 246
However, even if he couldn't take a physical form again, which he could, the below quote indicates that he could use and carry the One Ring whilst in his spirit form anyway.
But 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. There he took up again his great Ring in Barad-dûr, and dwelt there, dark and silent, until he wrought himself a new guise, an image of malice and hatred made visible; and the Eye of Sauron the Terrible few could endure.
The Silmarillion Akallabeth - The Downfall of Númenor