I can't speak to rifles as I'm not what I feel to be a proficient shooter of them yet. I can speak a little to pistols and what worked for me.
YMMV.
I deal with an interesting obstacle as I'm left-eye dominant, yet a right-handed shooter. The extra lean is something I don't wish to do either. Here's what I do to try and overcome, and yes, it's annoying at times, until I threw precision out, and focused instead on center-of-mass. CHL instructors, and other security types are saying that most SD situations are now beginning within 3-5yds. Precision is not quite the order of day if you train with that info in mind.
If you are going for precision, using the one-eye method is what comes natural. For SD, using two eyes is pretty important. Efficiency in that high stress is what will be the deciding factor of the outcome, the practice required to attain that efficiency will yield precision as a byproduct.
For SD, instead of using the sights at first try to aim using the gun's silhouette on paper. This is by no means an exercise in accuracy, as it is in just making sure the gun becomes an extension of your physical movement/objectives, without compromising your peripheral vision. It's safe to say that in a SD situation it'll be more than one shot getting off, and you probably won't have the time to line up your sights using a single eye to get that single shot you hope will resolve the situation. So to keep both eyes open, and aim, aim with the X in mind, but your goal is to hit within COM consistently.
Along with proper standing technique, grip, and trigger technique...
Practice, practice, practice, practice... the better groups will come on their own.