The public official, of course, is the president. Dowd is not a public official and cannot be bribed himself, but he could conspire with a public official to arrange bribes on the official’s behalf. The theory would be that Dowd and the president engaged in a conspiracy to accept bribes by agreeing that Dowd would make the offer. This, of course, would require proof that Dowd was acting with the president’s approval and not merely freelancing.
Neither bribery nor conspiracy requires that the underlying scheme be successful. The crime is the agreement itself, coupled with at least some steps to carry it out. If Dowd and the president agreed Dowd would offer an exchange of pardons for silence and he did so, that is a conspiracy to commit bribery. Whether the offer was accepted would not matter.
Thanks to the unusual circumstances in this case, bribery has a significant legal advantage over obstruction of justice.