I think they can use tissue and other such items. My poor body has been rung through the ringer. But whatever I can give, I'll give. And yes, you could have a donor card, be a donor on your driver's license, and your family could still say no. It's easier for the transplant people to approach a family if you have a card or it's on your license.
You can't take your organs with you to heaven. And heaven knows we need them here.