And finally, I think F instead of Am at the beginning and along all the second verse would be great!

Also, at the bridge, at the part when it says "I don't want you to run away so" I would strum an F before the G, and at the end, at "Will you hold me closer" I'd switch Am for F.

Instead of Am and C in the first two lines in the chorus, I would recommend F and Am respectively. It also goes well with the oooh's in the music!