I don't think the camp is full, I suspect the reservation outfit doesn't handle reservations for off season camping and so they just show all sites as full. The park is "open" for camping only April-Oct. While the MA energy and environmental affairs web site - http://www.mass.gov/eea/agencies/dcr/massparks/region-south/shawme-crowell-state-forest.html
- says off season camping is available, there is no indication on the site of how to access it. Maybe you just show up?
FYI, if you don't already know it, Sandwich is probably 4 -5 hours from NYC on a good day.
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