What is the best way to approach the rental agency about backing out of my Lease?

There is NO excuse for backing out. Read the lease and see what the penalties are for breaking the lease. Usually 3 to 6 months rent. Then that is what you pay. Deposits are nonrefundable in NY. Abandon the apartment then you will be sued. Your credit destroyed and the amount you owe increases do to interest, fines and court costs. Technically you are defaulting whether you move in or not.