A Face-palm for Sauron

[Inziladun]

We are all aware that the strategy of Gandalf and the Wise for defeating Sauron was dependant upon Sauron not realizing the Ring still existed and had been found. Gandalf explains it to Frodo:

'And this is the dreadful chance, Frodo. He believed that the One had perished; that the Elves had destroyed it, as should have been done.

I could question just how Gandalf could know Sauron's thoughts on the matter, but I'll let it pass for now.
The problem is, Sauron should have been well aware the Ring was still around.
After all, he was able to re-body after the Last Alliance. The Barad-dûr he was able to rebuild, for its foundations had endured.
One could argue that Sauron had no exact knowledge of what would happen were the Ring to be destoyed, but there is this, when Frodo and Sam stood at the Sammath Naur:

Then his wrath blazed in consuming flame, but his fear rose like a black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung.

So it seems he did have an idea what would happen to him upon the destruction of the One.
The Tale of Years says Sauron apparently first arose about the year 1100 of the Third Age and occupied Dol Guldur. Notably very near to the place where Isildur fell and lost the Ring in the Anduin. The Ring was found by Déagol around 2463.
Here's the issue: shouldn't Sauron have realized fairly quickly after he re-bodied that the Ring was still about? If he had, he surely would have discovered much more rapidly what had happened to it, and things would have gone much differently.
Does this make sense, or am I being too hard on the poor, second-rate Morgoth?