I think that the most important steps go before any sort of argument. They are:

• Get used to think logically and notice logical flaws. Including yours.
• Avoid vomiting certainty. (I call this “to be an assumer”.)

If you follow those two steps, you’ll be far more informed than most people out there, simply because you’ll make better use of the info that you’re exposed to.

Now, when you are in a protracted argument, here’s my “ideal” approach:

1. Ask myself “am I being irrational?”. If yes, I try to identify the source of the irrationality (e.g. emotional attachment). If I can’t get rid of it, I leave the discussion.
2. Ask myself “am I being an assumer?”. If yes, I inform myself further, then decide if I’m going to keep the argument or not. It’s fine to retract your claims, just do it clearly.
3. If the other side is being an irrational, assumer, or liar, I call it out while exposing why it matters. (There are polite ways to do this; I don’t bother.) The reaction of the other side dictates what I do: if they disengage I leave it be, if they admit it I might go on with the argument or not, if they turn into “lalala this hurts my fee fees then I claim that I didn’t read lol lmao haha” mode I usually mock it and disengage.
4. If I got tired of a discussion about something not too overly important, I simply leave. There’s no shame on that.

I say “ideal” because I myself sometimes don’t follow it, even knowing that I should. (I’m verbose and eager to dig deeper into any subject.)

Some key details here:

• Remember that in social media (including Lemmy) there’s always some sort of general audience, lurkers that are following the discussion but not interacting with it. What matters is less to convince the moron(s) and more to inform the general audience.
• Sometimes misinformation can be harmful, so there’s an actual reason to keep yourself in a protracted argument. Just don’t do it at the expense of too much of your time.
• I believe that it’s completely fine to answer blatant stupidity with rudeness, as long as your rudeness isn’t: misaimed at the wrong people, or in an environment with no tolerance towards rudeness, in a community that you hold a position of power (e.g. the discussion happens in a comm that you mod) or backed up by more stupidity. Your mileage may vary with that.