Lotteries!
I think giving a NY Lottery ticket to anybody who completes both vaccines would be a free and very effective for the state way to encourage people to get vaccinated.
If you give people a free lottery ticket it cost nothing more than the paper it’s printed on because of its unlikely to discourage regular gamblers from participating and buying lottery tickets. And even if it does discourage gamblers, it’s off budget and the losses would be temporary at worse.
BF Skinner used to talk about how effective lotteries are in encouraging compliance – people and rats respond stronger to a chance to win a large prize rather than be guaranteed a small prize. Lotteries to reward good behavior is a very powerful but under utilized tool.