The definition of I would be obliged in the dictionary is if you tell someone that you would be obliged or should be obliged if they would do something, you are telling them in a polite but firm way that you want them to do it.

For which I shall be highly obliged?

Kindly let me know when I can expect the connection. I shall be highly obliged if you take immediate steps in this regard. I would be highly obliged if you would take immediate steps in this regard.

What does it mean to say I am obliged?

much obliged in British English

or I am obliged to you. formal or old-fashioned. expressions used when one wants to indicate that one is very grateful for something. Much obliged for your assistance.