I think the main problem is the rushing for organs indeed. I don't know how this can be fixed, but maybe it should be harder to get death penalty. That way a judge can still rush it, but the criminal would need to have done pretty much to be convicted for death penalty anyways.
I think it should be so, that you give away your organs after death, unless you state you don't want to with valid reasons ( religion etc, not "because them people can't have my organs" ). Then instead of executing through organ removal, the ansthesis should be lethal ( because if you have valid reasons for not giving away some of your organs, you'll live for a small time after some organs are removed, which is not ethically good).
Though I live in Holland and we don't have the death penatly here, I think it's a pretty good idea. If someone is bound to die and you can save others with it, why not try to save other humans?