Can't really comment on Elkmont. My wife and I camped there about 15 years ago and it was OK. Like most any NP campground but nice enough.
As for fishing alone, I do it most of the time I'm in the Smokys (3-4 time a year unfortunately). Sometimes I'm with my son.
Unless you're going waaay back, the risk is minimal. Getting lost on the stream is not a big risk. Follow it downstream and sooner or later you'll run into your car or another road (at least with most of the streams in the Little River and WPLP watershed). Can't speak about the NC side and I think some of them do end up in a lake somewhere.
I think the worst risk is getting hurt and how long it would take to find you. When I'm there by myself, I do one of two things. I either call my wife back home and let her know what stream I'm going to be fishing (sometimes an email) or I write a note about where I'm going to be and put it under the windshield wiper on my car. I know that increases the risk of someone breaking in, but I rarely have anything of value in my car and what is there is insured. If I were really back up some creek with a broken leg or worse, at least the rangers would have a good idea where to start looking when they found my car.