I think you are playing this song in a key that is too high for your vocal range. If you try tuning the guitar (and everything else) down a half step or two you may find this tune is much easier to sing. I had to do the same thing with "Escape Velocity" which was originally written in E however I just couldn’t get the high notes. So, I played it in D and .....no more flat spots !

Hope this helps Dude
S