I'm also in New York, and our experiences were similar. Since January, I've booked about 10 appointments for various eligible loved ones: I'll bet each one took a solid 2 hours of effort: clicking, refreshing, waiting in online queues for multiple pharmacies and the NYS sites, etc. I cannot imagine what a nightmare this must be for those who aren't very computer-literate, or who don't have a great deal of time on their hands to keep refreshing websites in multiple windows all day.
It was worth it, though: by the end of this month, everybody close to me except my kids (ages 12 and 14) and brother-in-law (who is eligible, but inexplicably believes himself to be invincible) will be fully vaccinated.