Appointments book very quickly. When they get a free one,.they post it on a regular schedule, posting new openings every 90 minutes. Check their web page for appointments at 10:00am sharp, then try again at 11:30am sharp. Keep at it and you will get in. All shelters have the happy problem of too much demand and not enough supply. NEAS ships in rescues from far away (southern US typically), so they have more pets than most shelters in the area.
I believe it's about a week but holidays plus covid? I'd expect a bit more.
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