It's much easier to play opposite handed if you have a pre-shot routine and know what you are actually doing with your regular hand. Then, all you have to do is do the same thing with your opposite hand. It will feel a little uncomfortable at first, because it is new to the muscle memory. But, if you do things right or correctly, it will feel fine in no time.
^^^ This. Part of the ability to play well opposite-handed is having an established PSR that you can "mirror" (flip sideways) to the opposite site.
I like the snooker stance for this reason. It's very simple to "flip" to the opposite side, because all the checklist items are the same, and not dependent on "feeling comfortable." (This is not to say the snooker stance isn't comfortable, but rather that a proper snooker stance mandatorily places emphasis on where you place your feet, what direction are they pointed, locking the leg, where your weight distribution is, griphand arm alignment to the cue, head/eye alignment to the cue, etc. -- checklist items that with experience, become second nature.)
Because shooting opposite-handed is not "immediately comfortable" (hence why many of us don't like / don't have success with shooting that way), the snooker stance's rigid checklist of fundamentals is very helpful.
I personally like to shoot opposite-handed (lefty, for me), because I find it easy and reliable. I just go through my checklist, and I'm online -- I only need to concentrate on delivering my cue straight and correctly. And I often use it as a "spot" to play weaker players. In my Thursday 8-ball league, there's another player there that is equally adept at shooting opposite-handed. When he and I match up, for the purposes of boosting the competitive spirit, we'll sometimes choose to shoot the entire match opposite-handed. What's funny -- something that we both try our best not to let it on that we find it funny -- is when folks are watching our match, they'll notice something's wrong, but quite can't put their finger on it. Almost like they're watching pool from a mirror, but don't know it. Then, someone will finally "get it," and go, "wtf are you guys doing?!?"
-Sean