I have practiced business law for a fair number of years, and have yet to see a successful domestication (for either a corp. or an LLC). I have not shepardized your statute (reviewed the statute to see if there is any case law interpreting the statute to support domestication - keep in mind, the statute you cite is also the primary enabling statute for creating an LLC traditionally - so I am unsure whether there is any actual support for the use of this to transfer your company as a "domestication").
While I have not seen an LLC with a domestication, I have encountered corporations that have attempted this. The problem is that you cannot simply avoid the fact that the entity is a creation of state law, and you must ensure 2 things (1) that your business is in good standing (you appear to have some regulatory issues which are outstanding, I don't know what you are working with, but I would suggest working with a lawyer to help you sort these out before trying to wind down your current business - either with a traditional wind down, or a wind down and domestication); and, (2), you must continue to have someone act as an agent for the entity in the original state even after you wind it down (again, you started a business entity there under NY law, you cannot simply walk away from NY).
If you really do want to try a domestication (which I am skeptical of its success), please hire a local attorney. It may work out, but please do not do this without counsel (you may pay a little bit in fees, but doing it right the first time will save you a huge amount when compared to how much it will cost you to try to resolve it).
Here is the best summary on moving a company (LLC or Corp.) from one state to another. As you will see, the options I outlined initially are the ones provided here, and they are the ones that are utilized by most businesses: http://www.nolo.com/legal-encyclopedia/how-move-your-llc-another-state.html