This might work.... Use the polygons to clip the lines. Keep only those lines inside the polygons. Convert the polygons to lines and merge the clipped line sections to this new file. Convert the merged lines back to polygons.
I showed a technique for doing this at this year's user conference. It's on the model and script tool gallery - Fundamentals of GIS - Overlay . Download and unzip. In the folder "PointLinePolygon" is the plp toolbox and w/in that, there's a model called something like "Split Polygons". Open plp.mxd and run the model. It uses Feature To Polygon (ArcInfo only) followed by Identity (also ArcInfo only) to bring the attributes across. I'll have to think about how to do this w/o an ArcInfo license.