Dont use drape lines for that purpose - I would use the change elevation command and use the by surface option - this will elevate the lines to the surface but they dont become a dependent object on the surface like the drape lines do.
If you use drape lines then you have to explode them using the Explode command to break the link with the surface which makes them normal lines and then you can do whatever you want with them
The change elevation command allows you to also put in an offset to the surface as well while you are in the same process which is quicker as well
Hope this helps
Alan