The tragedy is that design moment for the column has increased after applying member offset to the column for the same load case.
Yes you can.
Can I apply member offset to the column and command the software to design the column for the moment at the bottom face of the beam ?
They will be different if you have specified a member offset. The moments reported in such a case - for the member with the offset - is not at the joint but at the face of the member, which is at the offset distance from the joint.
It can also be different if a plate or solid element is connected to that joint in addition to the column and beam. The unbalanced moment is the moment in the element at that location.
If you are referring to the moment used in concrete design, check whether it is reported at the joint, or at some distance away from the joint.
Regarding your question : The moment at the end of column is not the same as that at the start of the column of upper storey. Can this be manually adjusted?
If there are beams or slabs or shear walls framing into that column, the column below is not guaranteed to have the same moment as the column above at that joint. If you want them to be the same, you need to release the moments in all the entities framing into that joint except the columns.
At all junctions of beam and column , the sum of all the end moments in a plane say XY or ZY must be zero, as per the established law of equilibrium. In the staad output as also the design end moments for the beams and columns are observed to be different for the same load case. Why it is so? Some where I remember to have studied that staad does only one round of moment distribution. Can that be the reason? If it is so then can we manually balance the moment and design the beam and column as per the balanced moment . Generally the column end moment works out to be more than beam end moment. What is the reason for the unbalanced moment.. One round of moment distribution does not appeal to me. The moment at the end of column is not the same as that at the start of the column of upper storey. Can this be manually adjusted?