NYC gov Cybersecurity Job Search

Hello all. I am a native Brooklynite, Cybersecurity grad student, already have SOC experience at a NYC authority, and Security+ certified. I've applied to a couple of city/state infosec analyst roles months ago and it has just been total radio silence since. No status updates in the application portal at all. Anyone who is looking for cyber jobs in the NYC government running into this issue as well? I'm curious:

• How long is "normal" before first contact?
• Does cold-emailing HR ever help, or just annoy people?
• Does an internal referral actually speed things up

Any assistance is appreciated because I am also in the Cybercorps SFS and need a cybersecurity government job to fulfill my obligations and prefer to stay in NY.