Advil raises body temperature, so that won't help. All NSAIDS do, as they're fever fighters. I have this problem when I wake up, though the cause for me is antidepressants. It's particularly bad in the summer. This can be hormonal for both sexes. Have you tried taking a shower before going to bed? That's what I do, and sometimes it helps.
When you experience this, are you FEELING anxious, jittery? Is your mind on a lot of different things? Are you worrying about certain things?
Sometimes, some people experience anxiety like you describe because once you've lied down to go to sleep, you're no longer distracted with your day, therefore, anxious thoughts tend to pop up at that time.
Are you actually falling asleep? I would be careful about taking sleep remedies, even OTC ones, as they can end up compounding the problem, especially down the line.
I use to have this problem.I bet this is due to hight tension.When I was epxeriencing it my mind wanted to sleep but my body didn't.It helped me to go to bed every night at the same time...preparing my self for sleep...and never doing stuff before sleep.I use to just sit down and see a movie or something but nothing very exciting.The most important part is to go to bed every night at the same time.