You certainly can make the request. Further, if you are correct that the apartment was an illegal conversion, you might have grounds to break the lease. But before you go much further, I suggest that you retain experienced landlord-tenant counsel to evaluate whether, in fact, the apartment is an illegal conversion. I have found over the years that many tenants in your situation are mistaken about matters such as this. There is no substitute for retaining counsel to represent you in a matter such as this.
Dear Can i break my lease before i move in if I find out that the apartment is an illegal conversion? can i request another unit?
I will try to help you. If you were correct that the apartment is illegal, and you do not move in, you would have a defense to assert against any claim by the landlord that you breached your lease. In New York City, residential premises are legal to rent if the apartment has a certificate of occupancy. You could be certain of the violation by checking with the New York City Buildings Department at http://www.nyc.gov/html/dob/html/home/home.shtml.
However if you do disavow the lease, only you and the landlord can agree to another lease for another apartment. Your issue is not whether you can request the exchange, rather the issue is how this landlord will react to your refusal to move into the apartment.
If the landlord provided the signed lease and the keys to you, you now have legal possession of the apartment.
The landlord might claim a breach of the lease if you do not move into the apartment.
If you did not receive the keys, and the lease was not returned to you signed by the landlord, request your money back and find a legal landlord to start a relation with.
You should consider a consultation with a lawyer.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.