Interesting you say that - I'm also prescribed melatonin... but I chose to mostly skip it to stay productive i.e I only take it when I /need/ to go to sleep because I need to get up early for something the next day.
But for me, 5mg does tend to knock me out and I can barely stay awake after an hour of taking it.
What's interesting is my daughter has similar symptoms as you and has been diagnosed with Delayed Sleep Phase Disorder (DSPD). She's recently been prescribed a dose of melatonin which she takes an hour before the time she wants to go to sleep.
That appears to have helped somewhat.