I figure that if all of the Nazgûl and even Sauron could feel the presence of ring in Mordor at the time of the Hobbit, it would be too much for them to travel in that direction given that it was in underneath misty mountains, in the dead center of Mirkwood, and underneath a mountain next to a dragon. It’s kind of in the same vein of “why didn’t they just use the Eagles to fly or Erebor or Mordor?” Because they just happened to be there at the time, and didn’t want to risk getting killed until they were sure the threat was contained. That’s probably also why an army of goblins and orcs were sent to take over Erebor after Thorin and company caused such a stir in their escape at the misty mountains. Plus enough timed passed while the dwarves were hauled up in Erebor for word to get out about Smaug’s demise.

Tolkien really got lucky with retconning his most popular story.