Those roads do not go through any national parks. Be aware of the Blue Ridge Parkway and 441 from Cherokee to Gatlinburg, they are in National parks. The "road to nowhere" out of Bryson also goes into the park. If you go into the park, unload the firearm and lock it up, preferably out of reach. If you go into a part of the park there will be a large sign announcing you are entering the park. Enjoy the trip, It's a beautiful area to visit, I used to live in Sylva.
If you get into Nantahala Gorge stop and eat at Rivers End restaurant at the Nantahala Outdoor Center. Be warned they do serve alcohol and it is a hippie hang out. But great food.
Watch out for that dragon, I hear she bites occasionally!