I've fished from the Sugarlands up to the headwater streams of the WPLP, and I have never seen or caught a brown either. I think the LR is the best place on the TN side of the park for them, and like Rog and Speckleman said, the NC side is silly with them.
If you can stand the hike, Campsite #53 on Deep Creek is a great place to catch a slam all in one area. There's lots of browns, rainbows and brookies all within pretty close proximity to each other.
"Even a fish wouldn't get into trouble if he kept his mouth shut."