Nobody verified the birth certificate. That was not part of the process.
What was part of the process was that the chairman of the party had to swear an affidavit (under penalty of perjury) that the nominee was constitutionally eligible (which would of course include being a "natural born citizen").
The chair of the DNC was Nancy Pelosi.
I remember back then, as well, that the declarations in every state used exactly the same wording – EXCEPT for Hawaii.