I have enjoyed a half-dozen or so trips to the park, but entirely from Gatlinburg westward. I have no knowledge of streams on the east side. Still, within the park, the streams are not stocked. Nearly every stream with year-round water holds a decent population of wild trout, brown, rainbow and brook trout. The brookies are at higher elevations, the brown at the lower elevations and rainbows throughout.
Find the nearest park entrance that doesn't take you through a tourist trap area. Then, find the largest watershed and once you are in the park, surveil the stream as you travel "up." Most tributaries of the main stem in the watershed will be fishable. Areas that move away from the roadways tend to be better, but there are some great spots, just off the roadways.
I can recommend ways to explore if you have specific questions. According to my Google search, Cosby is north and east from Gatlinburg. If this is not correct and it is nearer to the Gatlinburg or Townsend entrances, I may have more specific advice.
Here is a great map for park fishing:
Rather Large Smoky Mountain NP Trail map PDF file
I would personally consider studying ahead on the Little Pigeon River drainage. The main stem in the park and all the various sub-basins.
Also, if you fish exclusively within the park you only need a basic TN or NC fishing license-- either state on both sides of the park, as long as you are within the park boundaries. The short term NC license was cheaper last time I checked.