One way I like to handle serious situations - especially if they can go negative - is putting things in writing - especially email. Written interactions are, well, documented and one can go back later on and reread what was said instead of trying to remember it.
Besides that, though, I find I can much better express myself sometimes if I can take the time to craft what I'm saying. I can type something, see how it looks on the screen, try something different, move this paragraph this way, split this other paragraph and so forth. I can also take the time to read what the other person has said and figure out which interpretations make sense.
This especially helps because (1) we Aspies take time to process things and (2) we're better off when we can focus on one thing such as writing, as opposed to multiple things at once such as words, tone of voice, gestures, body language, etc.
Unlike, say, with many tests, the first response is often not the correct one, and it can take a little thought to discern just what the other person really meant.
I've learned to understand people, and in turn make myself better understood, face to face. And especially in heated situations with people I know well, we can cool things down sometimes by switching to email.