I disagree with a few things being said on here.
I'm a right handed left-eye dominant shooter (opposite of you). I started shooting guns and bows right-handed, then I switched to left-handed at the advice of my local bow shop tech, and now I practice ambidextrous. It's been an expensive ordeal selling right-handed guns, buying left-handed ones, and then selling those for ambidextrous right-handed ones, but it's finally paid off. Let me explain.
First, shooting, IMHO, is not so complex as to prohibit someone from switching.
Although I had switched to left-handed shotguns, bolt-action rifles, and bows years ago, I always remained a right-handed pistol shooter slightly turning my head to use my dominant left eye (Hickok45 and Nutnfancy do the same thing). One day, I was dry practicing with my Glock 19 left handed and I noticed that, never really practicing drawing with my "off" hand, I was much more comfortable gripping it left handed. It didn't make sense to me because 1) I was right handed, and 2) I was shooting a pistol by extending it in front of me more like the isosceles stance (which supposedly doesn't favor one side or the other).
I had asked James Yeager about this previously when working on my drawing technique. He instructs people to bring their gun up under their eye and then pressing it out forward. I told him I was left-eye dominant, but he insisted it didn't matter (that I was shooting right-handed). People compensate by turning their head slightly (to use their left eye). No biggie, so that's what I did. In other words, as most right-handed shooters do, I pressed the gun forward ever so slightly to the right side of center (as to come up under my right eye if I didn't turn my head).
For me, however, I tried doing it left-handed coming up under my left eye looking straight ahead in an isosceles stance. That is, it looks like I have my gun perfectly centered in front of me, but it's actually an inch or two towards my left (lining up under my left eye).
You would think an inch or two would make absolutely no difference in the mechanics of my hands, and maybe it doesn't, perhaps it's psychological, but all I know is that I prefer it. James Yeager also teaches people to grip their pistol with more force from their off hand. This might have something to do with it because I'm stronger in my right hand than my left, so I naturally apply more force to my right (off) hand when I'm shooting lefty. This is an advantage of cross dominance IMHO. For you, holding your pistol right handed will allow your left hand to exert more natural pressure to improve you're shooting (if James is right).
It may have been easier for me because I also had the advantage of learning to shoot lefty shotguns and rifles previously. That made it even easier, I'm sure, but not to the degree that I couldn't make it up with practice in a relatively short time.
Today I just buy right-handed shotguns and rifles because they're easier to sell and trade. I don't mind the brass flying across my face like I once did shooting long guns, and it matters much less with a pistol. My long guns always have a tang safety, however, so I can operate them comfortably from either side. I'm not claiming to be a great shooter, but I am just as good shooting right or left handed.
If you practice and you still favor your left hand, remain left-handed/right eye dominant and turn your head. For me, it just feels better shooting left handed now, but I still shoot right handed much of the time becausewellI'm right-handed...it's not hard!
The only thing that's frustrating is finding left-hand friendly firearms. Mossbergs and Brownings are great for shotguns, Savage is great for rifles, FNH is great for pistols (but they only have full-size options at the moment). I can bow hunt white tail deer for over four months a year in my state, so I switched from using a left-hand compound bow to an ambidextrous Excalibur crossbow. Once I find a set of the perfect ambi pistols, life will be good!
Anyway, I hope this helps. It may not be as easy for other people as it was for me, but I absolutely encourage people to learn to shoot everything orientated with their dominant eye. If they put a modest effort and it's not comfortable, so be it, but I found the transition very easy to make. It will be even easier for you because finding right-handed equipment obviously isn't hard to do!