This law is ridiculous and I followed an effort to repeal it a few years ago (it failed), but there are two ways to work around it: you can use a registered age

This law is ridiculous and I followed an effort to repeal it a few years ago (it failed), but there are two ways to work around it: you can use a registered agent for your business in a county where it's cheap to publish (I believe for around $150/year you can register with an agent in Albany and advertise in a nearby newspaper for $75 or so), or you can delay actually doing the publication as there's no penalty, and only publicize when you absolutely need to (e.g. if you're suing someone). Also, if you're not wedded to the simplicity of an LLC you can start an S-Corp in NY state inexpensively and with no publishing requirement–though with more overhead and differences in structure.