If it is indeed a custom first person pronoun, they are using the wrong third person version of the verb.

Correct.

Dragon says they're using 'drag' as a pronoun, but they actually are not.

They are using it as a nickname whereby they constantly refer to themself in the third person.

They don't believe this is what they are doing, but it is what they are doing.

Its not actually a coherent neopronoun structure that works within the grammatical rules of English.

Further, I should add that Dragon states that their preffered 'drag' pronoun also operates in second person, on their profile.

Meaning that if you were talking directly to Dragon and said:

You like blueberries, right?'

...they would presumably view the use of 'You' as not respecting their preferred pronouns, which should instead be 'Drag'.