If you go "OK fine, I will incorporate elsewhere, like Delaware", then NYC views you as a foreign investor and mandates you still do the news paper thing.
Here is the kicker though, the news paper you are allowed to do this in vary based on county. You live in queens? That's a solid few thousand dollars. Brooklyn? Also a few thousand. Staten island? Less than $300.
If you do not do this then the enforcement mechanism is that when you get sued, then a court can take back the corporate veil perk of your llc (llc is void so it turns into a sole proprietorship I think) and go after your personal assets if you loose a lawsuit.