And if I haven't heard from them after more than six weeks passed?
Then you should contact them at the phone number or address provided on the New York State Board of Elections website.
And what will DMV do with my data?
You authorize DMV to send the Board of Elections your digital signature we have on file from when DMV issued you your Driver License, Non Driver ID Card or Learner's Permit. Agreeing to the attestation is the legal equivalent of a handwritten signature.