To prevent multiple edges between two nodes from overlapping, either ...
run "Tools" -> "Route Parallel Edges", or
select the edges in turn and change their port coordinates (i.e. properties "Source Port X", "Source Port Y", "Target Port X", and "Target Port Y"), or
enable orthogonal edge editing (third tool bar button from the right), select one of the edges and drag it up/down (if a horizontal edge) or to the left/to the right (if a vertical edge). This will only work for completely horizontal or completely vertical edges, though.
Alternatively, do not create overlapping edges right from the start. If you press and hold SHIFT before starting edge creation, the newly created edge will start at the mouse cursor position instead of the node center (and later end at the mouse cursor position instead of the node center).