Edmltw, I think you are more comfortable singing in chinese. Try doing away with reading and playing, learn the song well. Personally, reading and trying to play and sing adds up to 3 things. We need to translate in the mind what we read and again express what we read outwardly. Normally that means we can't concentrate 100% on singing and playing.
OK some musicians can do that well, but most beginners not.
Put a bit more in between piano when you are not singing.
Basically, you need to know the song damn well and feel really comfortable when you perform. When we feel that we still have to keep up or struggle in a song, that means we are still not comfortable with it. Only when we are comfortable can we fully express ourselves.
Feeling the meaning of the words can bring out more emotions.
Anyway, in 'Fly me to the Moon' there is too much sustain on the piano.
I suggest listening to more covered versions of this song by other world renown singers to get more ideas.
I hope this helps.