The DOE just posted high school info and handbooks online. Not sure the links below will work so I’m adding them again. SHSAT dates are listed in the back of the specialized handbook. A quick word about those: you don’t get a choice when you take the test. There are early and later sittings, Saturday and Sunday offerings. Those are determined by where your child goes to middle school, if they have an IEP or testing accommodations, religious observances. The specialized HS handbook also has practice tests in it, based on the new exam.
HS resources page:
I’d suggest signing up for their email alerts. They can be helpful.