I too have cross firing issues and I've tried all the solutions, such as the patch, without much success. First it felt uncomfortable and also my non dominant eye got tired after a while. For firearms there is a new solution out there, called XD Solution http://xdsolution.com. It helped me and I hope it will help someone else, too.
I shoot a lot of pistol, 40k rounds a year (at least). I am R handed and L eye dominate, so much in fact, that I bat/swing a Golf club left handed but do everything else right handed. Everyone's dominance is different but my solution is to shoot both eyes open using my right eye. It took awhile but now it is perfect. I will index the pistol and it is right there on my right eye every time. This was my solution and is the best for me, everyone is different, there is no one answer. It really isn't a hindrance unless you make it one.
I am left eye dominant and right handed. I do everything right handed except shoot bows and guns. I probabally cant even write my name left handed but I have shot plenty animals left handed. I made the switch when I was 17 years old and never looked back (I am 30 now). I actually like holding the foregrip of a rifle or riser of a bow with my dominant hand, to me it feels more steady that way. If i picked up a bow or rifle and tried to hold it right handed it feels totally awkward, which was not the case when i switched 13 years ago. It took some getting used to but it should not be too hard to train yourself to do it and eventually it will feel natural. I shoot right handed bolts and shotguns and doesn't really bother me (except I perfer to have a thumb safety on top of the stock, which is common with Browning rifles and shotguns). The only finacial commitment you would have to make would be to switch out your bow.
I would encourage you to try left handed. I know others said to stay with what your are comfortable with, which I cant say I totally disagree with (especially in the middle of hunting season). I can guarentee that switching to left handed won't feel comfortable for the 1st year, but eventually you won't know the difference.