It says in the Fellowship of the Ring that Sauron learned that the One Ring was still in existence, partially from Gollum. Gandalf says during his explanation to Frodo that "he believed the one had perished; that the elves had destroyed it,". However how could he have thought this if he was reincarnated. I thought that after Isildur defeated Sauron he was reduced "back into shadow" and that the only reason he was able to reappear was because the ring still existed. So wouldn't Sauron have known the ring was still around because otherwise he wouldn't have came back?
The Ring, and the portion of Sauron's power within it, is indeed what tied him to Middle-Earth, but there is no reason he had to be aware of this. If Sauron had information from a source he believed reliable that the One Ring had been destroyed, he would be likely to believe that his continued existence was due to keeping a greater part of his power in himself than he had previously thought.
No, this is not correct.
He learnt from Gollum that he once had a ring –
He deduced that Gollum's ring was the One Ring.
That's entirely different to learning that the One Ring was still in existence.
If the One Ring had been destroyed then Sauron would have been destroyed with it. The fact that Sauron wasn't destroyed means that the One Ring was still in existence (and Sauron knew it still existed) but didn't know where it was or who possessed it.
It's at this point that Gandalf suggests in The Fellowship of the Ring that Sauron thinks the One Ring is destroyed. Notice there's almost 2,400 years between the One Ring being lost and Déagol finding it in the River Gladden.
Notice that Sméagol/Gollum possesses the One Ring for almost 500 years before Bilbo finds it. Sauron is fully aware of its existence during this time.
Notice how Bilbo only has the One Ring for about 60 years.