Which documents do I need to switch an NY license over to an NJ license?

I did this. You need your birth certificate & if your married, a marriage license showing name change for a woman, your social, your ny id, a CC, lease & a bill with current address

Proof of address and NY i.d I'm pretty sure

What documents do I need for registration?

There is a six point system you need. If you go to tje njdmv web site it will give you the requirements. Passport, utility bill with new address, social security card are some examples. But they do have different categories so best to review specifics on their website