For example, if I live in District 1, can I only vote for candidates in District 1?
Yes, this. I've never voted in New Jersey, but in most places, your entire precinct will be in the same district and only able to vote in the races that apply to it. You won't even see district 2 candidates on the ballot.
Only one district per voter is a matter of federal law. You can only belong in one district and only vote in that district. There used to be at-large districts such that when there were two, your vote could count in both. However, this was used to prevent black representation, as a state that was majority white could allow the white majority to elect all the representatives. So the law was changed such that each person can only vote in one district.