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Mullica Hill is part of the Harrison Township school district and the Clearview Regional High School system. Of the areas you mentioned, this is the better school system, but all are "good".
East Greenwich and Woolwich are both part of the same regional high school, Kingsway. Kingsway is a good school, but not as highly rated as Clearview.
East Greenwich has it's own elementary system and the kids transfer to Kingsway starting in middle school.
Woolwich is part of a combined district with Swedesboro for their elementary system and the kids transition to Kingsway in middle school.
East Greenwich and Woolwich are pretty similar as far as ratings go, but I would personally give the nod to East Greenwich.
Another local option you didn't mention is Logan, which is where I live. The elementary education is rated higher than East Greenwich and Woolwich and equal to Mullica Hill, but it is part of Kingsway Regional for high school.