The Pseudodragon's Sting attack says:
the target must succeed on a DC 11 Constitution saving throw or become poisoned for 1 hour.
It's worth noting that no additional save is permitted.
Furthermore:
If the saving throw fails by 5 or more, the target falls unconscious for the same duration.
There is no mention of being Poisoned and indeed, it seems like they indicate this as a different effect that just runs for the same duration.
Being unconscious in combat can definitely be bad, but I think there are a lot of instances where being Poisoned for an hour with no additional save is worse.
Is this interpretation correct, or are you also Poisoned for 1 hour if you fail the save by 5 or more?
Compare this to the language used by the Ghost's Horrifying Visage:
…succeed on a DC 13 Wisdom saving throw or be frightened for 1 minute. If the save fails by 5 or more, the target also ages 1d4 × 10 years.
And the drow's poisoned crossbow:
…the target must succeed on a DC 13 Constitution saving throw or be poisoned for 1 hour. If the saving throw fails by 5 or more, the target is also unconscious while poisoned in this way.
The exclusion of any language linking the latter effect to the former makes it seem like Pseudodragons follow slightly different rules.
Best Answer
You are definitely Poisoned
As PJRZ pointed out, failing the saving throw by 5 or more is an example of failing the saving throw. As such, you will both be unconscious (because you failed the save by more than 5), and poisoned (because you failed the saving throw at all).
It is very reasonable for you to ask why some other game features would include the poisoned condition in the unconscious one, such as the Drow crossbow, which states (MM, p 128, bold added)
The distinction here is that features defined like the Drow crossbow require the target to be poisoned to be unconscious, while the Pseudodragon's sting does not. Anything which cures or prevents the poisoned condition will cure or prevent the unconsciousness condition created by its poisoned crossbow. RAW, the same is not true of the unconsciousness caused by the pseudodragon's attack.
For example, if a 10th level Monk was shot by a Drow's poisoned crossbow, and failed his saving throw by more than 5, he would not be unconscious because his Purity of Body feature means he was never poisoned (and is only unconscious "while poisoned"). However, if he was to be stung by a Pseudodragon and fail the save by more than 5, he would lose consciousness (though he would not be poisoned).