I am right handed but left eye dominant. I aim with my left eye. I think it’s called cross-dominance.
I have been told by random people I have good aim. I shoot well, I make golf puts, but I’m very average at darts and pool.
If I close my left eye, and use my right eye, would I be a better darts player?
I have cross-eye dominance, as well: right-handed, but left-eye-dominant.
Back when I golfed, it did seem that I was able to putt more accurately if I closed my right (non-dominant) eye. The downside is that you lose depth perception at that point, but in darts, when you’re essentially directly in line with the board, that might not matter.
Might be worth trying, though I might suggest closing your non-dominant (right) eye. However, the issue may be that throwing a dart is a different action than shooting a gun or putting, and not something you’re naturally good at, rather than a matter of aim.
Your stance while throwing darts may also be part of the issue: are you standing with both feet at the line, or with your right foot on the line (assuming you are throwing right-handed), and your left foot further back?
A lot of successful baseball hitters had their dominant eye as the one closest to the pitcher, and if they don’t they often will adapt an open stance to compensate.
Which also may be a factor in the number of players whose batting “hand” is different from their throwing “hand.”
I learned that I was cross-dominant when I was in high school (and took archery lessons). However, I always batted right-handed in baseball (and later, softball), and was bad at hitting. My grad-school roommate, who was a baseball player, suggested that I try batting lefty; I was able to learn how to do so very quickly, and after some practice, was able to hit better lefty than I ever had been able to righty.
It’s a very common condition and except for certain types of activities (i.e. keeping a full field of view in tactical shooting) not a problem.
It’s worth trying but it is less likely to make a difference than just being coached in throwing technique.
Stranger
I’m like the OP (and the rest of you). I’m also a good shot with a rifle, but until I calibrate it my shots tend to veer pretty far to the left, by up to a hand at 25 meters.
The one time that I ever fired a shotgun, it was when I was at an offsite conference for my first job, and they had a set of recreational activities one afternoon. Swimming, golf, and hiking didn’t appeal to me, so I signed up to go skeet-shooting.
Knowing that my left eye is my dominant eye, I talked with the instructor who was leading the session, and he showed me how to fire the shotgun lefty. However, it was still a standard shotgun, which ejected the shells out of the right side of the barrel – it was disconcerting to see the shells flash past the right side of my face. 
I close left eye with rifle, pistol is not hard to align to your dominant eye though I have a tendency to close. When I got a bow I just bought a left handed one. Shotgun is the tricky one because you’re “supposed” to keep both eyes open and I never committed to learning lefty.