Try cycling through the polygon features, making a layer of each polygon in turn by using a selection query (say, one feature ID or other unique identifier at a time). Do the polygon to line conversion on that layer, and then, after adding appropriate attributes to identify or label the lines, append the output to a collective feature class with all the line outputs. You can then blow away that single polygon layer and make a new layer from the next polygon to do the process over again for that one. (Rinse, repeat)
That process isolates each polygon during the conversion, so you avoid extra intersections; and isolates the output line so you can explicitly set the attribute / label; and finally collects all the lines in a single feaure class so it is easy to handle.
With lines features that (as Arc/Info used to call it) maintain seperate planer graphs (meaning they cross without intersecting (think highway bridges, or power lines crossing a road), you don't need the overhead of routes.