To answer your question I will first explain some aspects of a long distance move:
Long distance movers usually offer one month of free storage as a bonus for their customers. This "bonus" actually has a hidden economical reasoning behind it. Usually, when a long distance shipment is picked up it is delivered to a facility called "transitional storage", usually owned or rented by the company that does the pick up, or sometimes the transportation company that will deliver your items. Either way your shipment will be stored there until its turn comes to get on a long distance truck and be on its way.
Shipments don't get on the trucks that quickly, and keeping a shipment in the storage for a month (and sometimes even longer) is a common thing in the moving industry. In the summer the hauling space gets scarce, and shipments have to wait long to get hauled. In the winter the shipments are scarce, and again there is a delay because trucks are waiting to be loaded. This way a free storage is a fact of business, so offering it movers usually don't loose anything, but have consumers convinced they got some added value to the deal. Not a bad thing, but still good to know.
In your case a move from Georgia to New York has some advantages. Because this route has many trucks returning half-empty to New York from Florida you can get a good deal on free storage. If you choose this scenario I would recommend you to negotiate at least 2 months of free storage.
Second scenario is a bit more problematic. Basically, your shipment would be picked up, stored in the transitional storage, then hauled to New York where it will be unloaded in another storage facility (that you choose), and when you are ready you will have to hire local New York mover to deliver it from the storage to your new apartment. This scenario is more expensive and increases the damage to your items as a result of extra load-unload cycle.
Having said all that, I would like to warn you about transitional storage. Those facilities usually are not very aesthetic places. The items are often stored on wooden pallets on the floor, shrink-wrapped to keep them together. Cases of lost (or stolen) boxes, or forgotten items and sometimes even pallets are not rare. The facilities are often not climate-controlled, and storing furniture or other household goods for too long in such environment is just not recommended.
To protect yourself from all that I would strongly recommend you to visit the transitional storage where your shipment would be stored and inspect it religiously. Insist on having your items stored in a separate storage room if possible, and on the day of the move come with the movers to the facility and oversee the unloading and storing.
I hope all that information will help you make the right decision.Good luck with your move,
Alex Z. @ MovingAnswers.org Follow us on Twitter for valuable tips on making your move easier!