From your original story, it doesn't look like you have noticed that your cached belief was floating. Presumably it's a one-off event for you, and the next time you feel frustrated like that, you will know what to look for.

Now, I am not a rationalist (IANAR?), I just sort of hang out here for fun, so I am probably not the best person to ask about methodology. That said, one of the approaches I have seen here and liked is steelmanning the opposing argument to the point where you can state it better than the the person you are arguing with. Then you can examine it without the need to "win" (now it's your argument, not theirs) and separate the parts that work from those which don't. And, in my experience, there is a grain of truth in almost every argument, so it's rarely a wasted effort.