If conjugation depends on the individual pronoun, the pronoun you used was third person since that’s the conjugation you used. They/them is also third person and singular and plural are conjugated the same, so the comparison doesn’t apply.
To be clear, please use any and all pronouns you’re comfortable with. But don’t write a third person sentence and insist it’s first person.
Conjugation very much builds grammatical structure and is not independent from it. But that’s not what I said. As you said, conjugation depends (among others) on the pronoun. You said you were using a first person pronoun but then used a third person verb. This does not add up. You can either say “drag am always punctual” to make a first person sentence or “drag is always punctual” to make a third person sentence.