I wrote two times before about organ donation in various aspects here and here. So yes, you would know that I am thoroughly for organ donation. And here is the issue in today's newspaper. Basically the question is, if you are dead, can the hospital use your internal bits for use in medical research or for saving somebody else's eyes or liver or what have you?
Most people do not care, they just go through life unknowingly and just like that. While organ donor wait lists (see here for a heart-wrenching list in the USA, 100,000 people waiting for an organ while only 6000 transplants were done).
I realise there are issues like people stealing body parts like this, or improperly storing body parts. But that does not remove the need for organs. Now, the main problem that I see is that the surviving heirs and relatives are unable or unwilling to let those organs go. Or you yourself are unwilling to let them go. And plus you do not trust the government to deal with resources. The answer is simple, allow people to sell their organs.
Either before or after, just like you should be allowed to prostitute your body just like you do your mind freely, your organs should be open for sale as well. In many cases, such as in India, Brazil and other poor countries, that is what happens. So you end up with a market that exists but is illegal.
Just like prostitution, you need to make this legal and allow people to buy and less under regulated circumstances, so that this huge imbalance between supply and demand can be removed. If you think that there is a problem with only rich people taking up organs, then well, have an organ bank by compulsorily harvesting organs from the dead and giving the heirs due compensation just like we do for cases around Eminent Domain or Compulsory Purchase.