I too am right handed and left eye dominant ... but have no problem with it. The only thing I shoot left handed naturally is sling shot; don't really know why that is esp. when I shoot my bow right handed.
What I have learned from this thread is how different everyone is in regards to this. I grew up shooting lefty (RH, LED) and couldn't go back with rifle or shotgun without a lot of practice and wouldn't do it unless I had to.
Some others are the same way, others not so much. So my advice is to whatever feels most comfortable for you. Comfort with a gun = relaxed = repeatable, which is what it is all about.