Buying a used car with no credit and in a different state?

Please let me provide a little background:

I just moved to New Jersey from California.

My license is currently in California.

I am young (21) and have absolutely no credit at this point in time.

I am looking to buy a used car which will at the maximum be $5000.

If the car does cost the maximum specified I will need to finance a large portion of it (Currently I can put down $500 to $750)

The question I have is I do not have any proof of residency in New Jersey yet.

My paycheck stub is still linked to my California Address.

I also do not have a utility bill or any bills linked to NJ yet
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That being said,
Since I am going to need to finance will need I need a proof of residency for NJ?

Is there a way around it?

What can I do?