there was an Eye in that tower that did not sleep
This does not necessarily mean an eye was floating around the tower, could it not refer to the fact that as a
Maiar,
Sauron did not require sleep, and therefore was always alert. It's no different to me saying to my children 'I've got my eye on you'. A metaphor. I also recall vaguely a comment that Sauron had many ears, but it never occurred to me to take this literally.
Tolkien could have said in that sentence that
Frodo 'felt the focus of Sauron's mind searching mentally for him', but it certainly wouldn't have been so gripping as the image of the eye searching. Plus if you take that quote completely literally, you would have to admit that Sauron also had a finger, which in turn implies a hand, arm and the rest of a human body. Just because all that is seen in Visions is
Sauron's eye, does not mean that there is nothing else there.
I take your point that Sauron could have taken any shape he wanted, and that it is not specifically specified which shape he had taken during the
war of the ring. But as his normal incarnate form over the millennia had been that of a man, (the Sil states also that when the
Ainur took mortal appearance they were humanoid, if elven instead of human), it seems reasonable to me to assume that this is the form that he took once more. After all, he could have appeared as a spotted tadpole if he wanted, but it wouldn't have helped much. I'm honestly not mocking here, but it seems only logical to me that he would have chosen the most practical form available, and if he wanted to wear
the ring, he would need a hand.
Practically speaking as well, if Sauron was incarnated as a man when
Isildur cut the ring from his finger, and his power was so much diminished by that maiming, it also occurs to me that he may have been unable to take any other form until he regained the ring, I mean we already know that he was unable to take 'fair' form again after being caught in the downfall of
Numenor, so why not a further limitation when his power was so drastically curtailed? Just a random thought there, nothing to back it up I'm afraid.