This 1-parameter family requires the solution of a 6-dimensional period problem and the effort yields a 1-parameter family.
One limit decomposes into a doubly periodic Scherk surface and a doubly periodic Karcher-Scherk surface, glued together end-to-end with catenoidal stitches. This is already complicated, but the other limit is truly spectacular:
We see here singly periodic, 8-ended Scherk surfaces with an eccentric handle. That something like this exists follows from Kevin Li’s unpublished thesis, where such surfaces are constructed near limits when all Scherk ends become horizontal. Below are the flat structures of G dh and 1/G dh on a simply connected reflectional fundamental piece.