I exercise if I can't sleep. Which is almost every night lol.
I start by going for a run around the neighborhood. If that's not enough I do sit ups or lift weights till I finally fall asleep. For getting back to sleep? More of the same exercises. It works for me. Just work my body till I can't go any longer and just fall into my bed and wake up X hours later.
As for early morning games? I like JRPG's, something about waking up and staring down a hundred hour long monster feels great lol. Right now i'm working on replaying Dragon Quest 9 and Disgaea D2. ...that ones going to take a lot of early mornings.
As many others have said, podcasts have been an absolute lifesaver for me. I'm a pretty anxious person and my mind tends to wander if I don't keep it occupied. A chilled out podcast puts me to sleep every time, to the point now where I'd probably struggle to go to sleep without one on in the background! No wonder I'm single.
I have pretty bad insomnia on and off. I usually fall asleep listening to podcasts. If I wake up and can’t fall back to sleep lately I’ve been watching old Game Room quick looks. Those usually make me pretty sleepy. Or I read. A night shift filter and blue light filtering glasses have seemed to also help with the phone not contributing to the sleep problems.
Melatonin only works, at least for me when you are tired and close to falling asleep. It's not meant to be a help you fall asleep but help you sleep properly. I have been dealing with sleep issues for more than half my life and I'm in my mid 30s here. For me there is no one cure all method, some things work for me like melatonin, but then it just doesn't. Sometimes just listing to music or focusing on something repetitive to helps my mind relax and puts me to sleep.
Do try and get a sleep study done, often they are free and you can actually get help as to why you are having issues falling asleep and then you can maybe find a doctor who is actually able to help understand what is the actual cause of why you have problems sleeping.