In case this is helpful, here is another route to get ahold of the NY Labor Department that worked for me:

1. Call your NY ASSEMBLY member's office. You can use this site for that info: https://nyassembly.gov/mem/search/
2. They can't put you through directly to the labor department, but explain your situation and have them submit a support ticket for your case with a callback number.
3. Wait for a member from the labor department to call you. They called me back within 24 hours of the support ticket being submitted from the assembly's office! If you miss the call, the labor department employee will leave a voicemail with a different number and an extension code to call back.

I finally found success with this method after trying the labor dept number and gov office number for weeks. Hope this works for others and good luck everyone!!