Hi Louise, What you do depends on the purpose of your map.
Does the blue line accurately display the stream on the ground? Is it something you will be using to make a land use decision? If yes, then delete the pink line where it is adjacent to the blue line. You can do this by making a clip file of the database reference file with a map extent you make ( a rectangle polygon a bit bigger than your map). That way you can edit it without altering the reference geodatabase. You can then append your clipped flowline with the stream shapefile and make it one big happy family.
I put NHD flowlines on my maps all the time. They are a good indicator that there may be something on the ground, but they need to be checked out in person. Streams move and meander. Sometimes the map lines themselves are moved by accident in GIS.
Best to walk the streams, especially the small ones, both seasonal and perennial, and make tracks with your gps using field maps. You'd be surprised how different they can be from what's on the database.
