Like others said, play with the dosage. First time I tried it, I had 2.5mg (or maybe 3mg I forget) tablets. I had a somewhat similar experience, I'd get drowsy, but when in bed I'd feel cold and couldn't sleep very well at all.
The next year I got 0.1mg tablets, experimented a bit and found that 8 of those (0.8mg) was the sweet spot for me (taken about 2 hours before I want to sleep). I generally used it in the winter, when it gets dark at 16:00, which apparently can throw my cycle a little out of whack if I don't pay attention, the melatonin seemed to help to fix this a bit.
I have to say, I've never noticed really strong effects like "puts me out quickly". Just a general feeling after an hour or two, that "yeah I could sleep". If I listen to that feeling, I sleep very well. If I stay up anyway, it goes away again, and I don't really get the benefits.
One thing I noticed, after a few weeks, was when I knocked over and broke a glass of water for the third time. I wondered, "hey that seems to happen quite a lot lately..". And always in the evening after I had taken the melatonin, so I guess it does do something to my coordination or perhaps alertness. This was easily solved by taking a bit more care where I place my drinking glass after taking melatonin :)