Interesting Marc, the last part of the surname -shin I had trouble finding. The first part of the surname Dyach I think perhaps derived from Greek diakonus, also as Latin diaconus, meaning "servant" also as English form "deacon", a servant perhaps a priest or a cleric. If so, the meaning of the Ukrainian surname Dyachishin would be "deacon's wife" or "son of the deacon's wife", the servant or deacon perhaps being a priest or cleric.
There are also the Scottish surname Diack and the Polish surname Diak/Dyak/Dyjak, perhaps they are also of the same origin.