Shooting with both eyes open is beneficial for all of the reasons stated above. If you feel uncomfortable doing it, you’re probably over thinking it. For some, shooting with both eyes open can’t be done due to some sort of a physiological reason. For most, it’s a matter of what we’re used to and what is natural.
Next time you try it, try not to think about it and just do it. If you’ve been shooting for a while, just let your eyes and body do what they’ve been trained to do and keep your head out of it. Your brain is going to want that crystal clear two plane sight picture. Tell your brain to shut up because you know what you’re doing.
Sorry I can’t be more specific, but this was something that I struggled with years ago and really didn’t get over it until I started shooting matches when closing one eye meant you might run into something or miss half of what was going on. Now, I still close one eye when I’m shooting at ,say, a target at 100 yards with a handgun, but doing it isn’t automatic any longer. I now leave both eyes open. I just quit thinking about it.
That tip by atldave was true. Try shooting with a dot sight. Heck, put one on your handgun. Lots of temporary setups available these days.