Sometimes when I wake up I have a song in my head complete with all parts as if its real.

I heard that Prince wrote The Ballad of Dorothy Parker after a dream, and in the SOTT film he is often seen dreaming/napping..most notably before and after U got the look. Does this imply that he dreamt that song too?

I also heard that Sir Paul McCartney wrote Yesterday after dreaming it.

He was apparently napping when writing Little Red Corvette. He would wake up write a line or 2, and continue that cycle.