@Cotty: Thanks for your reply! Perhaps I'm missing something in there, but in using your solution, I simply end up hiding the lines rather than fixing the geometry.
@Gaieus: I'd never used the sandbox tools before; I'm quickly starting to realize how useful they can be. Unfortunately, it seems like some of the more critical lines cannot be targeted by "Flip Edges." Is there a reason for this? I seem to get so close to fixing my problem with this.
@Peter: Going through that thread helped me understand the importance of proper triangulation. Embarrassing to admit, but I had never known about the "show hidden geometry" option before now. I can properly triangulate the space between my curves when they're on a flat Z plane, but I run into a problem. When I try to raise one end of the shape after it's been triangulated, it only raises the very last lines and the immediately connected triangles. How can I smoothly slope the whole shape? Also, as far as realism goes, this is for a project that isn't steeped in real architecture, so a little bit of silliness is allowed.