"Blackmail" isn't usually the wording used in statutes. Statutes generally refer to coercion or extortion.
In Texas, Rick Perry (who at the time was Governor), threatened to withhold 7.5 million dollars in state funding if a particular official did not resign. (Wikipedia, Dallas News)
This is still in the middle of litigation, but an appeals court found a charge of coercion of a public official was improper. A charge of improper use of his office is proceeding to trial. (New York Times)
"Extortion" charges are generally related to actually seeking money, property, or services.
For example, former Illinois Governor, Rob Blagojevich was convicted of extortion (New York Times, Chicago Tribune). Extortion charges were related to Blagojevich holding up a grant for a school in congressman's district until that congressman held a fundraiser for Blagoevich. He also threatened to hold up hospital funding until that hospital's CEO made a donation to his campaign.