You are correct in you will best be set with a local landlord who will understand your situation. You will get bonus points for being a teacher, they are generally awesome tenants.

You MAY be able to find larger places in Rahway, Cranford, etc, who take bonding agreements. Basically you pay a 3rd party a fee (usually about a month's rent), who has an agreement with the property that they will cover your rent if something goes south. (note it doesn't get you off the hook for rent, the bonding company will now be the ones who chase you down), but it can help someone without a history, perfect credit, or with an unusual job situation at the time they need to sign a lease.

The student loan stuff is BS, nobody cares as long as your income is in line with the 30 or 40x gross annual income to rent rule and your debt to income ratio is healthy.

One thing you may want to do is reach out to your new school district, and see if they know places that have been cool with renting to teachers in the past. I wouldn't frame it as, "I'm having a hard time" i'd frame it more as "I'm looking at a few different places, do you guys have any recomendations about good spots for teachers to start out from?"