As an individual sitting at home, I would keep the same opinion that I hold now: it is immoral. However, as Governor the issue would be even more difficult for me. In a time of such racial tension, failure to sign their death sentences would be seen as being soft on racism, even if it was cited with best intentions.
That is where the real difficulty would be for me.
Illini,
That is a very interesting hypothetical, and now I will propose my second question for all of you:
Now imagine that you are the Governor of the state in question and the judge has handed you the death sentences. A practicing Catholic, you have strong reservations about the death penalty and strongly believe in the right of life. However, you have for all of your life held progressive views on race (though you keep those to yourself, for fear of being outed before you gain office), been opposed to Jim Crow, and don't want to send a wrong message to people that lynching is alright.
The sentences are on your desk, do you sign them? If so why? If not, why?
Provided that my Lieutenant Governor is also progressive on race, I'd resign and let him sign the sentence. Yeah, I'm a coward who hates responsibility.