If Sauron exits throughout the Hobbit, why does the Nazgul not try to seek the ring from Bilbo?

Great question! And, as (almost) always, amazingly, Tolkien provides us with an answer!

The TL;DR is: Sauron and the Nazgul were looking for the Ring, they just had absolutely no idea where it was.

Sauron could sort of sense the Ring, but Tolkien tells us that the Ring grants power according to the measure of the bearer. Frodo would have been a very small, probably unnoticeable "blip" on Sauron's radar, even when he had the Ring on (which is why Gandalf knew that giving the task of destroying it to Frodo meant it had the greatest chance of success, since Frodo could sneak into Mordor essentially unnoticed.) One other thing to say about "giving power according to measure": It is often repeated in the Hobbit and LOTR that Hobbits have an exceptional talent for not being noticed. One could even say that this was the Hobbits' "power." So if the Ring augments the bearer's power, then it would be reasonable to assume that the Ring made Bilbo/Frodo super undetectable (also partly the reason why Hobbits disappear when they put on the Ring).

All that aside, the minute Sauron learned of the Ring's existence, he threw a great portion of his power into looking for it. But how did he find out?

Gollum.

After the tricksy little Bagginses stole his precious, Gollum didn't just slink back into the mountain and cry himself to sleep, he went after it! And Gollum is more shrewd than we give him credit for. he knew vaguely what the Ring was, or at least had some idea of the evilness of its power, so much so that he eventually decided to go to Mordor to try to get it back, since he somehow felt that the Ring was headed there, or wanted to go there.

He followed the creeping call of the Ring and its master until he was eventually captured by the orcs. They tortured him, and eventually he told them that he had had his precious stolen from him by Baggins, who lived in the Shire.

That was all Sauron needed. He sent his Nazgul out on their long chase, and Frodo's great flight began.

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