The real question is "why Alabama?"
But if you are validly formed in another state, then you simply register the existing foreign entity to do business in NY. You aren't creating a new LLC, only registering the existing southern belle.
There must be a reason why you want a foreign LLC. There are tax sourcing rules, personal jurisdiction rules etc, and none will detract from your being primarily taxed based upon your state domicile. I don't know why you created a Bama LLC and not a NY state LLC. Maybe its to try and separate your location from the business site. Beware the jurisdictional issues: If Bamallc hires a secretary in New York, it will be doing business in New York as an unlicensed undomesticated entity and will have to file for permission (and pay additional franchise tax) to do business in NYC.
Thus, I have found the main consideration is whether you will have "feet on the ground" in another jurisdiction without being licensed or domesticated to do business there.
Same comes up her in California all the time. People Love NV LLC's, but if there are ANY "feet on the ground" activities in California, there is no liability protection in California. CA courts get to skewer the orchestrator of CA activity. So make sure that the BAMAllc never does activities in NY that would subject it to the NY jurisdiction of the NY courts.-- You would be exposed!.
Curt Harrington Patent & Tax Law Attorney Certified Tax Specialist by the California Board of Legal Specialization PATENTAX.COM This communication is general information and not legal advice, and does not create an attorney-client relationship. This communication should not be relied upon as any type of legal advice. Please note that no attorney-client relationship exists between the sender and the recipient of this message in the absence of either (1) a signed fee contract and (2) remission of an agreed-upon retainer. Absent such an agreement and retainer, I am not engaged by you as an attorney, nor is any other member of my law firm.