Needing help moving just the lines of a drawing.

Create a white shape. Select > Selext Same > Fill color. Delete

Create a shape with stroke and fill of None. Select > Select Same > Stroke and fill. Delete.

Should do it.

default1z,

How about repeating the Image Trace, only applying Ignore White for this second purpose?

The problem with using Image Trace is that the results it produces are a mass of compound paths which can be very cumbersome—if not impossible— to work with—even for an experienced user. I've found very little rhyme or reason for the way it sets up the order of the shapes. I would only recommend using the feature for an image that you will not need to edit. The best way that I could suggest for your current need is to use your result as a locked template and manually create the linework by drawing over it with the pen tool.