I'm also a left-eye, right-hand shooter. My advice:
Try a few different things and find what works for you!
Everyone is a little different on this subject, and there is no one-size-fits-all solution. All of the ideas mentioned above are tried and true, as are a few others.
For me, I started shooting when I was 5 years old (I'm 35 now). I didn't know the first thing about eye dominance at a young age, and I've always had good vision (it's just better in my left eye). I learned to shoot most comfortably as a right-handed shooter, with rifles, pistols, and shotguns. We live in a world where most of these things are set up for right-handed folks.
As such, since I'm a right handed person living in a right-handed world, I decided to continue shooting right-handed, even after discovering eye dominance issues when shooting pistol. For me, the problem is most obvious when I'm shooting in low light environments, and while shooting pistol. For rifle it's easy enough to shoot with the non-sighting eye closed or squinted, and you can still shoot very well this way (despite what some might tell you on this subject).
At times I've found myself needing to blink or squint my dominant eye when taking longer range pistol shots using a right-hand, right-eye stance. It probably creates a small handicap in my shooting, but it's not a huge one at all (I'm in LE, and shoot pistol better than most of my coworkers — maybe I won't make GM in USPSA, but I shoot well enough this way that I don't see a lot of value in trying to switch to lefty at this point).
As for rifles, how I shoot them depends on how the rifle is set up:
1) Under high magnification scopes: close the left eye, use the right eye. I've never seen any advantage in having both eyes open while shooting through high magnification optics, since each eye's image is essentially on a different plane of focus. Keeping both eyes open can reduce some eye fatigue, but it's not been an issue for me (and I shoot a lot of precision rifle).
2) Under no magnification (open sights) I just close my left eye and shoot with my right eye.
3) At work, where I want to maintain a high level of situational awareness while shooting my rifle, I run an Eotech holographic sight on my patrol rifle. Even with eye dominance issues you should be able to shoot through these optics with both eyes open… at the very least I can, and most other cross-dominant shooters I know can as well.
Anyway, that's how I do things. As with anything, there's the "right way", the "wrong way", and "may way". Your milage may vary. My spouse has this same eye dominance issue, but she's a newer shooter who learned of eye dominance the first time she shot. I don't think she's entirely settled on a method of shooting yet, but she's currently using a right-handed pistol stance, with her head tilted to allow her left eye to sight the pistol. She's learning rifle as a left handed shooter so far.
So, each person may choose a different path, and I'd simply suggest finding what works for YOU!
EDITED TO ADD:
Even as an LEO I've always run factory sights on my pistols. I just switched my primary duty weapon to using TruGlo fiber-optic/tritium sights, with yellow rear sights (dimmer) and a green front sight (brighter). I haven't used these sights extensively yet, but so far they seem easier for me to "find" quickly with my non-dominant right eye, while keeping my left eye open.