As to your first paragraph, one of the more liberating things about internalizing the Sequences and inculcating myself with Less Wrong memes is that it I've come to hold very few opinions strongly. When you move yourself to a sufficiently morally-relativistic framework you stop identifying with your opinions. Instead of saying there are things you "just have to believe" I would say there are things that it is instrumentally rational to behave as if you believe.
Regarding your second paragraph: I find that I simply get into fewer arguments. Because I have let go of most of the opinions that typically weight people and make them respond at an emotional level to being contradicted, every conversation that includes a disagreement becomes a joint truth-seeking venture. Instead of arguments, I have "discussions." If the other party in the discussion is not interested in truth seeking but is instead interested in being right, I just stop humoring them and change the subject. If they are someone I can be honest with, I will point out that they seem to have an irrational bias regarding the topic in question.
It seems like your first mistake was getting involved in a theological debate. Science is flawed, but "religion" doesn't even have enough predictive power to be falsified. I would step back at least one level and urge you to ask yourself what your objectives are in participating in such a confused discussion in the first place. I myself have indulged in totally stupid internet arguments which I can only attribute to a sort of perverse pique that strikes me at times, but I generally admit that I'm already failing to be rational just by participating in such.
When you move yourself to a sufficiently morally-relativistic framework you stop identifying with your opinions.
"Stop identifying with your opinions" is a classic Less Wrong idea, but moral relativism is not.
I don't think anyone here is perfect, though a lot here is simply brilliant logic. Though, as humans, we're automatically biased for or against something, at times for the wrong things and reasons. No bias at all isn't something we're equipped for, and to me sounds quite dull. Some things we have to believe, even as with our limited lifespan it would be very hard to learn everything.
Anyway, on to my actual point: How often do you realise you're biased for the wrong reasons, and you argue more for winning than being right? Also, what irrationality can you point out, and how to fight it? This could apply to you, or those around you, doesn't matter as no one's perfect. But as usual, we should try to at least be better.
Right now I'm trying to figure out how to best deal with unreasonable demands. A theological debate which eventually went "Well, if science can't do X, it is flawed, and we should accept that some questions are unanswerable.", and I'm working out the most efficient counter-argument (leaning towards "that's why we should try to see if it's impossible or not"). Anyway, share your thoughts on this, how to be efficiently rational, though if you try to help me, I'd like more questions instead of answers. I value the process of rationality more than straight-up answers, even though it's frustrating as it is.
Basically: Flaws you encounter and how to fight them efficiently, maybe [Bayesian Judo style](http://lesswrong.com/lw/i5/bayesian_judo/).