Andrew, if you get tired in your eyes by shooting with both eyes open, you are doing something wrong. Are you trying to focus on everything at the same time, or what? Remember, focus on front sight.
siotwo, yes, you will see two barrels, but if you are truely dominant on the eye you're using to aim with, you will be able to ignore "the extra".
I just realised what the problem might be. When I, and those who knows this stuff way better then me, say to focus on front sight, that doesn't mean that you should LOOK at your front sight with both eyes, and still be able to actually aim. If you try to do that, you will get really messed up.
Instead, what you do is (sort of) this, look with both eyes on your target, bring gun up so sights line up with your right eye, then change focus from target to front sight.
Hard to describe something that comes so natural to me, and I guess it's possible that some is simply unable to do it.