For someone in your situation - holding the SMC JSV who has landed a skilled job - the correct thing to do is to send it to the nearest INZ office, covering the area where you are now living. If in doubt, this
http://onlineservices.immigration.go...or.htm?level=1 will help. If you have sent evidence of your new job and your passport, normally they stick in the visa label and return the passport within a short time, normally only 2 - 4 weeks.
The adviser should certainly be able to tell you which office your application to get your Residence went to, and if any replies have come from that office in the meantime. But yes, you can ask INZ, if you aren't getting clear answers from your adviser.
As people have said above on the thread, Essential Skills visas are taking much longer than has been normal - just at the moment there seems to be a backlog. INZ have told some new applicants that it may take 4 - 6 months, but I think (just my guess here) that that is in the hope of making people be patient and not keep asking about progress, because some people on the forum have received their visas in around 2 - 3 months.
But you SHOULD NOT NEED the Essential Skills Visa. Your Residence should be through far more quickly than the end of May. Your original CO has already checked out all the evidence about YOU, and the Residence is waiting for you as soon as they can see you have a skilled job (which is just one thing to check).
Nobody applies for an Interim visa. It is something INZ automatically issue. But if it came to that, you would not get permission to work on it, because you are applying for a different sort of visa from the one you hold at the moment. The details are here.
https://www.immigration.govt.nz/new-...isa-conditions