I can do them fine, I just can't ever get them to the note or pitch I want them at. For a riff that uses pinch harmonic on the 5th fret A string (drop A tuning) I'm supposed to hit an A, but I can never hit it first try. I try to pick roughly around the same spot for it but my hand moves back after for the natural position.
Well, you can find more than one pinch harmonic on one note. You just change the place that your thumb hits the string. There are, as you will soon find, only certain notes that you can hit. In order to hit the notes outside of those ones, you have to bend the string.

Hope this helps.