I can't sleep well at home, so it's no different on a solo, lol. I too suffer from what you mention, I sometimes take the dog with me (she tends to do patrol all night), other times I actually like to camp by moving water since I know that things are ok and it helps me to sleep. On my last week long hike in the SNP (I still owe a trip report from last month), my worst animal encounter was with a dad-gum WHIPPORWHILL! He set up literally right over my head and wouldn't go away or stop....at about 2am I was up literally throwing rocks in the trees to try and run him off - the folks in the shelter heard the noise, saw my headlamp going up and down, and wondered if I was fighting a bear or something. I maybe got an hour of sleep all night, then walked 20 miles out of the park the next day. Not my finest hour.
All that is to say, as many others have, just keep doing it. You'll eventually get over the fear. My desire to be in the woods overcomes my fear of the dark every time.