New Jersey abolished slavery in 1776

New Jersey did not abolish slavery in 1776. That year, they adopted their first state constitution, which contained no bill of rights. You can read the original in full here. It remained in effect until a new constitution was passed in 1844.

As Henry Scofield Cooley noted in A Study of Slavery in New Jersey, because the 1776 constitution had no bill of rights, nor any other language ascribing natural rights to the state's citizens, the New Jersey courts would have been unlikely to have ever declared slavery illegal under the state constitution, even if the opportunity had arisen, which it never did. The earlier New Jersey laws, and British common law, remained in effect on the issue of slavery, which made it legal.

What the 1776 constitution did do was give equal voting rights to all free people in the state, as long as they were "worth fifty pounds proclamation money" minus any debt, for a period of twelve months before the election in which they were voting. Since there were free black people in New Jersey, they were allowed to vote just like any white person. Same went for women. All people who were free, including women and formerly enslaved people, all had the opportunity to vote if they'd saved up enough money. But this provision was later repealed, disenfranchising both white women and all non-white people.

New Jersey abolished slavery on July 4, 1804. Though, like many Northern states, they did it under a "gradual emancipation" law, so that there were still enslaved people in the state as late as 1846. (And, according to an article by Geneva Smith published on the Princeton University website, this actually went on a bit longer than that for at least some enslaved people.) While there were some petitions at the county level to repeal this law, it never happened. The 1804 law had been passed after a couple decades of lobbying by abolitionists; they had earlier succeeded in banning the importation of newly enslaved people from Africa in 1788. Once passed in 1804, the gradual emancipation provisions were adhered to. Male children born after 1804 to an enslaved parent were free at the age of 25; female children at the age of 21. Enslaved people born before that were still enslaved for life, until an 1846 law abolished slavery for the rest (though, again, as Smith's article points out, the language in the updated law still left a handful of people enslaved).

and vote against the Thirteenth Amendment

According to The Ragged Road to Abolition: Slavery and Freedom in New Jersey, 1775-1865 by James J. Gigantino, the reason the Thirteenth Amendment failed the first time around in New Jersey was entirely due to political partisanship. The Democrats controlled the state legislature in 1865, who viewed the amendment's ratification as a signal to the end of state power, a return to a strong federalist government, and "a roadblock to reunion" in the post-war period that could be a threat to the hard-fought peace between North and South.

However, this very issue became a campaign issue in statewide elections in November 1865. Republicans campaigned on it, and it helped sweep them into majorities in the New Jersey statehouses. The new Republican controlled legislature ratified the amendment on January 23, 1866, in the first month of the new term.

EDIT: Corrected link to the relevant page in Cooley's book.