It says in the fellowship of the ring that Sauron learned that the One Ring was still in existence, partially from Gollum.
No, this is not correct.
He learnt from Gollum that he once had a ring –
After the creature Gollum, who had previously possessed the ring, was captured, Sauron had him tortured and learned that he once had a magic ring, and from him he heard the words Shire and Baggins. He deduced that Gollum's ring was the One Ring, and sent his servants the Nine to find Shire and search for Baggins, so that the One Ring might be found and brought back to him.
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.
One Ring Timeline
Second Age 1500, Celebrimbor forges the 16 rings under the instruction of Annatar, who is Sauron in disguise.
Second Age 1600, Sauron forges the One Ring to rule all the others.
Second Age 1700, Gil-galad and Tar-Minastrir destroy Saurons army forcing him back to Mordor to regroup.
Second Age 3261, Ar-Pharazôn, the last of the Kings of Númenor, do battle with Sauron in contention of his self-proclaimed title of Overlord of Middle-Earth. Sauron is defeated and captured but through the use of the One Ring is able to turn the Númenóreans against Valar and towards worship of Melkor and human sacrifice.
Second Age 3429, Although Sauron's body is destroyed in the Fall of Númenor, his spirt was able to bear the One Ring back to middle earth and he wielded it in his renewed war against the Last Alliance of Elves and Men. The One Ring is cut from Saurons hand by Isilidur.
Third Age 2, Isildur loses the ring in the River Anduin just before he is killed by an orc ambush.
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.
Third Age 2430, Déagol finds the ring in the River Gladden, a tributary of the Anduin and is killed by Sméagol for it.
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.
Third Age 2941, Bilbo finds the One Ring.
Notice how Bilbo only has the One Ring for about 60 years.
Third Age 3001, Bilbo hands the One Ring to Frodo.
Third Age 3018, Sméagol dies in Mount Doom and the One Ring is destroyed.