I think CreeDo touched on it. While I have a better motion at the moment as a lefty, sort of like an artistic feel, shooting righty if it's not so spasmatic the shot feels much more deliberate. I dunno, I'll probably try to play for a couple of weeks righty outside of league and serious play.
I'm truly in the same boat you are. I'm naturally a right-handed person, with a strong tendency to be truly ambidextrous in many things. However, I have a eye condition that causes double-vision almost all the time, unless I cock my head in just the correct position.
While I've always written right-handed, I picked up a cue stick and gun to shoot left-handed from Day 1.
A long time ago, back in the day, it occurred to me what I was doing by shooting with "the wrong hand". I switched over to a righty one day and while it felt weird, I didn't lose much in performance. So, I started putting a bit more effort into it and now I can shoot either way and not really think about it.
My modest record is a 2-pack in 9B from both sides of the plate, of which I achieved as a righty first......on an Olhausen no less

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Anyway, I shoot primarily as a lefty, but never give it a second thought when it's time to switch over to a righty to get correct body position for a shot.
I've also noticed my stroke is much stronger by switching to my off-hand. I can draw the hell out of a ball and generally get more action. Now, there are times I get excited and scoop it for a miscue, but that's just me trying to do too much when I should just "trust it".
I've seriously thought about suffering for a year or so and playing as a righty, as I think I could actually reach a higher level. My vision is much clearer from that side and I don't experience any double-vision, as it puts my head in the correct position to avoid that malady. However, I know it would take a lot of work to get the "consistency" and muscle memory that I have now from the left side.
Obviously, I'd love to shoot at a high gear from both sides and I'm sure at some point I'll be more than proficient at that.
Keep us posted.....