I listened to 'My Life'.
Don't sell yourself short on your singing (as you do on one of your pages). You have a great voice that's reminiscent of Billy Joel himself. You also put the vibrato in the right places and so on.
Sometimes your accent is right on in terms of Billy Joel's own accent and sounds great, sometimes it sounds tentative. You could go either way but it's better not to stay in the middle -- either do the whole thing with a Billy Joel accent or do your own thing completely.
On that recording it sounds like you have everything solidly learned but still you yourself aren't really enjoying it. You have to enjoy it.
Otherwise -- and this is connected to the last point -- there are moments where the rhythm gets bunched up. You could practice with a drum machine to keep the tempo up and in agreement between your voice and piano.
Working on: A Night in Tunisia; Memories of Tomorrow (Keith Jarrett).
Just started: Brazilian Like (Michel Petrucciani)
Baldwin Model R (1974), Hardman 5'9" grand (1915), Rieger-Kloss 42.5" vertical