I think either they misunderstood you or you misunderstood them. A social security card/number does not get issued without you actually applying for it once you arrive in the US. I would suggest you visit the SSN office again and explain to them clearly that you have not yet applied for a SSN. Then, fill out a SSN application. Make sure you put your current address in that form. It does not matter what address you listed in your DV application form, they will send you your SSN at the address you provide in your SSN application form. They will also need to make a copy of your passport, so take your passport with you when you visit that office. SSN usually takes two-four weeks to arrive in the mail from the date you apply. I don't think it has anything to do with the type of visa you arrived in the US in, although they do verify your visa status with department of homeland security before issuing a SSN. Your Green Card will be sent to the 'other' address that you mentioned to them at your port of entry in NY. If you did not mention a different address during your entry in NY, it will be sent to the address in TX.