Marriage License

When does the bride and groom usually go for the marriage license?

The morning of the rehearsal dinner? Day before RD? Do you need to make an appointment?

You can go any time once you are 60 days from your wedding. You just have to wait 24 hours for it to become valid so don't wait until the day before. We went to the town hall in Hempstead about a month before. You did not need an appointment. It only took around 10-15 minutes. Here is link to info on getting your license from this website.
Marriage License

we went about 3weeks before

Here is a list of town halls in Nassau County and here is the list of town halls in Suffolk.

Here is the list of requirements to get a license. You need to get one within 60 days and 24 hours before the wedding.

You don't need an appointment, but some of them stop taking folks before the actual closing time because of the lead time to fill out the paperwork. Plus, their hours vary, both on regular days and near holidays. For example, we got our license the Friday before Easter (Good Friday), and some of the halls were closed or open only for a half day even though it's not a holiday.