My husband and I have similar personalities. We both avoid conflict, and very much dislike being upset or upsetting the other. We rarely argue and have never had a "full blown argument." We disagree, sure, but seek to find the truth of the matter over "being right."
I think, for myself, the first thing I consider is "could he be right? Is there something I'm not thinking of?" I try to shut down my emotional response and listen to what he is saying. My husband is a better listener than I, so he's usually already actively listening. :-)
I then restate what I think I hear. "So, I think you're saying (this, that, the-other). Is that what you mean?" And then I confirm. "Oh, so what you mean is....I like/appreciate/respect that." And restate or reframe my own statement. "What I was trying to say was..."
Usually we are on the same page, just communicating differently. It's a matter of reframing and coming to understand the heart of the matter in a loving and respectful way.
We come together in that we are both effective communicators who love one another and want the best for eachother. That's powerful. Often we embrace and thank eachother for being able to speak our minds and truths freely and kindly.
ETA: I also can't just let a disagreement be. I turn into a nervous wreck. The remnants of past codependency arise and I start in with catastrophic thinking. I feel like there will be terrible outcomes if we don't talk through whatever it is that we're hung up on and get past it. Silly, I know, but it sure does mean that we don't stay mad long!
ETA2: Nervy said, "giving each other the benefit of the doubt and not looking for an insult when one isn't intended." That's absolutely the truth!
I've noticed that many couples who have major arguments seem to have forgotten that their spouse is FOR them...and they are for their spouse too! We want what is best for eachother. I see spouses talking about one another as if they can't tie their own shoes without supervision...and it's no wonder, with that sort of attitude, that they fight. Love and respect are a must in any argument OR agreement, above all else.