I would agree with all that has been written above. As far as flies are concerned, I find myself using a handful of patterns - as has been said many times, presentation is far more important than pattern. That being said, the general trend is darker-colored flies early in the season, and lighter colors as the year wears on. For a dry, you really can't go wrong with a parachute Adams; it's highly visible, and it is sort of a "generic" pattern that matches a lot of different insects. In the summer, I switch over to the yellow PAs. As for nymphs, you really can't go wrong with the basics - Prince, Tellico, etc. I tend to "go small" on my fly selection - in other words, if the general trend is to fish a #14, I'll go with a #16 - I figure that a smaller fly will pick up fish that might be a bit skitterish, particularly if the water is low and clear.
As for a location - Pete brings up a good point about Eagle/Hazel...you're kind of trapped there if the weather gets bad. I've never fished that part of the park, so I can't say much about it. However, if you're looking for some solitude, and not too bad of a hike, I would suggest the Palmer Creek and Pretty Hollow Gap trails in the Cataloochee area; while the road into the Cataloochee valley isn't the greatest ( I was up there a few weeks ago in the snow, and decided to turn around when it started to stick to the road), it isn't bad in halfway decent weather (hard packed dirt near the park boundary), and you eventually get back to a paved road once you get to the valley. I've never camped there, but I have hiked a good ways up the Palmer Creek Trail, and it isn't bad at all. Backcountry campsite #39 is on the Pretty Hollow Gap trail, which branches off of the Palmer Creek trail. Definitely invest in a good park map - LRO has the National Geographic map, and it has a lot of detail on it.
Hopefully, we'll be up there the last week of March ourselves.