No, I'm one of those weird people that because my family moved to the UK when I was little in the late 80s for work for a year I'm under risk of mad cow disease and none of us can donate blood or organs. Learned that the sad way when trying to give blood in college, like half a dozen random things that can disqualify you that you might not realize.
You can donate your organs, if you die of a head injury. You cannot donate tissues like bone, skin, or corneas, because those are considered "life enhancing." Donated tissue has a similar restriction to donated blood, so you're right about that bit.
Organs like the kidneys, liver, heart, lung, pancreas, and stomach are "life saving" and don't have any restrictions. When you die, you'll be evaluated for any potential donation, and if you are a candidate for donating organs, someone will come talk to your family about it. Talk to your family about what you would want.