Never heard of adzuna.
The usual job sites for NZ are www.seek.co.nz
and Job Search | Indeed
Yes there are lots of people on LinkedIn but not sure if it's popular for securing work here ?
My personal opinion, but I wouldn't bother with recruiters. I'd look and find the jobs myself and apply myself.
I was signed up with a recruitment agency for a few months and they were a waste of time.
Recruiters are only in it for the money and the quicker they can get a person an interview/job offer/recruited the easier it is for them and the quicker they get their fee. If you don't have a visa that allows you to work here they'll place you at the bottom of the priority list. They'll look to find work for the people already here with the right to work first as those people are a better bet, will take less work to find them a job and cost them less money so they make more profit....Remember, the recruiters only get paid when a suitable candidate is introduced to the company, selected for interview, offered/accept a job.
If you do wish to use a recruiter it is easy enough to sign up online and/or via email but you'll have to constantly keep getting in touch with them to make sure they are working for you. Only sign up with one of them as NZ is a small place when it comes to jobs. Signing up with multiple recruiters has the potential for an employer to receive multiple job applications from the same person for the same job and they don't like it....apparently.