It can be related to where you grew up and where you are now. I was born in Chicago, where the nights were so bright you could see pretty clearly once your eyes adapted. Then we moved out to the Arizona desert, and on a moonless night it was incredibly dark. My folks gave me a night light without even asking whether I wanted it. It started out in my room but eventually migrated out to the hallway. I don't think I was ever "afraid" of the dark the way you describe but I was certainly "bothered" by it.
Then I moved to Los Angeles to go to college and it was rather distracting to have the nights be so bright again--difficult to fall asleep actually. Then after decades there my wife and I moved up into the redwood forest, where the trees block so much of the light that even under a full moon the house is so dark that I can get disoriented just finding my way out of the bathroom.
Fear of the dark can be irrational, but it can also be either caused or aggravated by circumstances. Some locations are much darker than others and they can make you feel helpless because you actually are a bit helpless. You could be at a tremendous disadvantage if a real emergency happens during the night. Every year a handful of people really do have snakes crawl into their house, burglars, tree branches falling through a window, earthquakes, etc.
I debated about putting that last sentence in but I decided it was worth taking the risk of making you even more uncomfortable in order to try to put a rational face on your fears. There are no monsters under the bed, no spiders descending from the ceiling, no bats trying to smother you--there really aren't and to still worry about them is to have some unresolved childhood issues that are beyond the scope of SciForums to deal with.
But there are real things that happen during the night and it's okay to worry about them. What's important is to put them in perspective, what we professionals call a "risk analysis." Consider that several hundred Americans are killed by lightning every year. Do you worry about being struck by lightning? Do you even follow the standard safety precautions in a thunderstorm? Probably not. Several thousand Americans are killed by drunk drivers every year. You probably don't worry about that either. This is a rational risk analysis. The odds of one of those victims being you is so low that you'll have a much greater impact on your longevity by putting your energy into learning about nutrition, doing exercises, and urging your friends and family to vote for sane politicians who won't get one-sixth of the world's population pissed off at us.
In this context of rational risk analysis, where does the risk lie of having something icky and dreadful happen to you in a dark house? Earthquake, serial killer, rabid bear, falling satellite, all of those risks combined? Those risks are real, but they are less likely than bee stings, which kill off several dozen of us every year.
Put it in a rational, statistical context like this and then see how you feel about it. If you're still worried and want to take precautions, that's fine because how you run your life is entirely up to you. But at least you will be running it rationally instead of based on a fear you can't quite understand.
The LCD on your radio is a good idea in any case. It's easy to trip and fall when you get up to go to the bathroom, find out why the dog is whining, or close the window because it's too cold. If one day you find yourself living in a dormitory room or a studio apartment, you'll have so many digital appliances surrounding you that you will be able to see rather clearly by their light. Especially things like VCRs with clocks, cable TV boxes that always show the channel number, etc.
These days most people have digital clock radios with LEDs that are always on. As you get older and you need a larger display so you can tell the time without putting your glasses on, you'll find that your bedroom is plenty bright enough at night.
Funny story related to this... I've been living away from home in Maryland for several years and it's taken me a long time to get used to life east of the Rockies. I had never seen a firefly before. My basement studio is crammed with furniture, but it's full of digital appliances. I've gotten used to navigating my way to the bathroom without bumping into anything or turning the light on and waking myself up too abruptly. One night I got lost. Kept stumbling into things, losing my balance, really odd. I simply could not find my way, even using all of the little LEDs, which come to think of it seemed unusually bright. I finally gave up and turned the light on--after first finding it because by now I had no idea where I even was. There were several fireflies that had sneaked in when I let the dogs out on last patrol, and they were sitting on the floor, on the ceiling, and other places, completely screwing up my navigation and even my sense of up and down. They look just like LEDs.