We've discussed this issue for four years, ever since the New Jersey indexes to births, marriages, and deaths appeared, but I keep seeing this problem on message boards.
This is the death certificate for Catherine Meeker (1795-1884), daughter of David Dean and Phoebe Beach. She died March 14, 1884, but the online index at Ancestry and FamilySearch has the year 1883 because the volume of deaths was July 1, 1883 - June 30, 1884. So 1883 was chosen as the year for all deaths in the index.
Most unfortunately in New Jersey, you cannot access original certificates of birth, marriage, and death online. You can access indexes online at New Jersey State Archives, Ancestry, and FamilySearch. Any index is only as good as its creator and does not replace the need for the original record.
For New Jersey certificates, you need to write to the Archives or visit in person (or have someone do it for you). The records are on rolls of microfilm and are searched by hand.