[The Might]

you are indeed right there: Sauron let Gollum go hoping that he will be able to follow him and found out where this "Shire" was...but Gollum fooled him

and here follow a few quotes all from the UT to clear up the situation:

[After his release from Mordor] Gollum soon disappeared into the Dead Marshes, where Sauron's emissaries could not or would not follow him. No other spies of Sauron could bring him any news.

According to Aragorn Gollum was taken at nightfall on February 1st. Hoping to escape detection by any of Sauron’s spies he drove Gollum through the north end of the Emyn Muil, and crossed Anduin just above SarnGebir. Driftwood was often cast up there on the shoals by the east shore, and binding Gollum to a log he swam across with him, and continued his journey north by tracks as westerly as he could find, through the skirts of Fangorn, and so over Limlight, then over Nimrodel and Silverlode through the eaves of Lórien, and then on, avoiding Moria and Dimrill Dale, over Gladden until he came near the Carrock. There he crossed Anduin again, with the help of the Beornings, and passed into the Forest. The whole journey, on foot, was not much short of nine hundred miles, and this Aragorn accomplished with weariness in fifty days, reaching Thranduil on the twenty-first of March." [Unfinished Tales: The Hunt for the Ring]

And here is what happens:

But when Sauron learned[size=-1] of Gollum's capture by his enemies the situation was drastically changed. When and how this happened cannot of course be known for certain.[/size]

But evidently later (since the lands of Thranduil would now be closely watched), possibly a month later, Sauron heard the
disquieting news that the Wise were aware of Gollum, and that Gandalf had
passed into Thranduil's realm.
Sauron must then have been filled with anger and alarm. He resolved
to use the Ringwraiths as soon as he could, for speed rather than secrecy
was now important. Hoping to alarm his enemies and disturb their counsels
with the fear of war (which he did not intend to make for some time), he
attacked Thranduil and Gondor at about the same time. He had these two additional objects: to capture or kill Gollum, or at least to deprive his
enemies of him;

So clearly, the order was to capture or kill Gollum. Only then, when need was at its greatest did Sauron eventually decide to send the Nazgul after Gollum - when all else failed. That is why I doubt he would have used any of them as messengers at Erebor.

In conclusion, I agree, we can only make attempts to find the right answer.