Are there actually no available DMV appointments in Manhattan through June, or is the website bugging?

Hi everyone, I’m trying to reserve an appointment at one of the DMVs in Manhattan through the website but all of the days through June, which is as far as the calendar goes, are grayed out. Is it likely that it’s actually fully booked, or is the website just busted?

In the case it is fully booked, am I just screwed, or can I show up and eventually get seen?