There is a theory that doctors won’t try as hard to save you if you’re an organ donor.

This is such a ludicrous theory that is sadly repeated often. It is extremely unlikely that your set of organs will be compatible with someone they know who needs organs. Even then, a nurse or allied health professional is much more likely to make a medical error which kills you. You'd be surprised at how thin the line between life and death is, especially for people on death's door.