I think you would have to do one or the other, but there is no reason you could not register for both.
Mum had cancer, which had spread into various organs and her bones, so she could not be an organ donor (she was 74 too).
I suppose it depends on the circumstances in which you pass away. If you have had a recent operation you cannot donate your body to science, but you might be able to donate some organs.
Register for both and let your family know of your wishes. Hopefully it is not something they have to act on for a long time.
With mum we had the conversation about medical donation many years before but only registered her 3 months before she passed away due to the sudden diagnosis of terminal cancer.
Mum's ashes have now been returned to us and she was scattered in Cornwall where she grew up.
Hopefully this thread has been useful and made people realise that just like donating your organs, donating your body after death can be so worthwhile for moving research and medicine forward.