You might try searching for prior posts about this, as it may be a common problem. This may very well be 100% incorrect, but it seems to me that adding the very slight lag from the time the audio leaves the machine and gets to your headphones, your response time, and the lag from your mic to the computer, would yield enough to create this effect. I haven't tried this, but it occurred to me, perhaps you could try adjusting the doubled part back a bit in time (probably on the order of milliseconds), to sync it back up with the original track. That then introduces phasing problems, etc etc.
even when I think I am singing the same exact part, upon playback it sounds like two people singing the exact same part, jsut a tiny bit off from one another.
