I think some people naturally hear harmonies and some hear the melody loud and clear. Sadly (from the point of view of singing harmonies) I am one of the latter. We have to really work at it. The best way I found was indeed to write it down and learn it as if it was the tune. Then sing your "tune" alongside a recording of the person singing the melody. Until you have it off perfect.

Then you will never be able to sing the melody again!!! But I can still sing "my" harmonies against other people singing the melody.

I'm still not very good at improvising. However often I try singing along in the car I still spring back to the meolody.

Helen (a different one!)