I was told that the Shastras do not recommend organ donation after death, as they do not really belong to us. Is organ donation right?

When we take the state of affairs today, we perform so many different deeds and activities against the Shastras towards our selfish ends and personal benefit. When it involves others, we try to find refuge in Shastras to discover a reason for not doing it.

When a deed is done for a social cause with the welfare of the people in mind, one can go ahead and do it.

Is organ donation after death acceptable according to our religion?

Organ donation is done as a social cause, with the welfare of someone in mind. So from this viewpoint, it is acceptable.