How should I stop thinking that I am even slightly enlightened?
To add another perspective to the existing answers (because you might want to know opposing or mutually-exclusive extremes in order to steer a middle way between them), maybe you shouldn't "stop thinking that you are even slightly enlightened".
For example if the Buddha and the Dhamma (the Triple Gem) cannot even slightly enlighten you, then what's the point of "Buddhism"?
There's at least one Pali sutta which says that conceit (that you can become enlightened) is helpful (because you hear that someone else has become enlightened and you think, "Why not me?").
And you might find it helpful to read this answer which describes Dzogchen and the earlier answer which describes tantrayana.
Saptha's answer said, "Only Supreme Buddha knows".
One way to stop thinking that "I am enlightened" might be to think "Buddha is enlightened. Am I Buddha?"
Various answers follow.
One is to consider Anatta. You wrote, "I become unaware of my mental formations" -- maybe it would be better instead to think that, "There are mental formations, but they are not mine and not me."
To avoid "identity view" you might also find it helpful to think, "There is (sometimes) enlightened practice, enlightened understanding -- that too is what it is and not mine or me."
I don't mean you should try to renounce "right view" or "ethical conduct" (by saying that they are "not me"), what I mean is that adding an "I am" view to those things is an extra complication which you and they may be better without (see also the answers to this topic which describe 'identity view' and 'conceit').
I have this constant irritating feeling that I am kind of enlightened
An alternate way to view that might be that the "constant irritating feeling" is of not being enlightened.
However maybe "I am not enlightened" is an ego-conceit, like "I am enlightened".
Also I think that the "irritating feeling" is a symptom of a fetter or a hindrance. Apparently you have associated "being enlightened" and "feeling irritated" -- maybe that's a mistake to associate them though, maybe it's better to associate "being enlightened" with "not feeling irritated" (and by "associate" maybe I mean saṃskāra i.e. "things put together").
If you do have any "irritated feeling" sometimes maybe that's not surprising! For example maybe it's uddhacca, or passion, which you might expect of/in anyone who's not an arahant.
Maybe not "constant" though, I guess it's more probably experienced as "continual" than "continuous":
Things that are unceasing or exist without interruption are continuous ... they never pause. Things that occur frequently or recur intermittently are continual. The continual action doesn’t happen ceaselessly, but it does happen regularly.
Another thing to note is that a characteristic of irritation is avoidance, i.e. if you characterize something as irritating then your inclination might be to avoid (not experience) it. Beware that this might be unwise, that it might be better to be willing to experience (better to know) your actual thoughts and feelings (I think that "be mindful" and not "be averse" is what some meditation teachers tell you).
A little more about this. ... It is definitely weird. ... It is funny too.
Perhaps, if you want, your relationships with other people (e.g. parents), and how you "check myself against enlightenment", could be the subject of some other questions.
And also I would like to know if there was any such situation (in the texts) which the Buddha faced.
Well, I'm not sure how that's relevant but MN 26 is one the suttas in which the Buddha describes his quest for enlightenment. I read it as saying that he successfully learned what his teachers were teaching (and didn't for example pretend that he couldn't learn what they were teaching).
He also had no "doubt" (see also vicikicchā) when he did finally attain enlightenment.