How can I get rid of that weird shadows for edges in edit mode with viewport Ambient Occlusion (Properties Panel > Shading) enabled?
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Because the way viewport's AO is currently built (2.77), everything is drawn in the 3D viewport is taken into account.
What you see is probably the projection of adjusted faces that comes from the unmodified mesh. Basically, as in edit mode you see two version of the same geometry (even with "Adjust cage ..." enabled, it's only projected upon the modified mesh), you get AO from both meshes.
If it is suitable for your workflow, one possible workaround could be to disable the Faces option in the Overlay section of the Mesh Display panel in the Property of the viewport.