The claim that a deceased person has no use for her organs, that the integrity of her body after her death is of no importance to her, is a claim that many dispute, typically in the context of some religious beliefs or others. It seems best for the state to avoid policies that some citizens find offensive on the basis of religious beliefs which the state is in no position to refute. Fortunately, we can avoid such policies in this case: by making organ donations opt-out, as you suggest, we'd have all the organs we might need.
The reason against this which you cite ("They're my organs and nobody else gets to decide what to do with them") is not a good reason against the opt-out solution. Yes, they are her organs and she alone gets to decide what to do with them -- but we still need to have a fall-back default for those cases in which a person dies without leaving clear instructions. Here any default society might settle upon is in the same boat, e.g. subject to the objection that it may not be what the deceased would have wanted when she was still alive. Still, society needs to set up some default and may then democratically select and implement any default so long as it is widely publicized and can easily be overruled by the person while she is still alive. In this democratic process, I for one would argue for the opt-out solution because it would bring huge benefits to people in need of organs.