The gender markers are unimportant in organ transplantation. The HLA system has to be well matched for most organ donations, this is the genetic system that is examined when 'tissue typing' is performed. If an organ with a mismatched tissue type is transplanted it can be quickly rejected as recognised as 'foreign' by the recipient . The alternative situation is when a person receives a bone marrow transplant, if the marrow graft is too dissimilar to the recipients HLA system, the bone marrow graft will reject the recipient in a horrible syndrome known as graft versus host disease (GVHD). Where the patients organs are systematically destroyed by the bone marrow graft resulting in a rather hideous death. These people are treated with anti-rejection drugs to try and overcome it.
I have several patients who have had bone marrow grafts from the opposite gender and now carry the opposite XY or XX gender marker than they were born with. BTW it does not change their gender or sex. Which may be of interest for discussion with people who suggest that if you are XY you have to be male or if XX you have to be female. Gender is between the ears not between the legs

I would encourage people to be organ donors and to make their wishes known to loved ones. The organs need to be retrieved shortly after death and it can be difficult for family and loved ones to make the decision when they are in deep grieving. Several lives can be saved by one donor. Two people can have their sight restored, two people can receive kidneys, usually one person or some times more can have a liver transplant, and one person can have a heart lung transplant, in the future pancreatic transplants will be successful and goddess knows what else.
You will be well remembered by many people for being an organ donor. Which I think is a nice thing for your loved ones to look over.
Hugs
Cindy