You can get cardboard box suitcases at the UPS store that are just like suitcases, only cardboard.
And places like uline.com sell boxes in pretty much every known dimension to mankind, though you may have to buy a dozen of them.
However, I would recommend something like a duffle bag, where you can get lots of stuff in them but they take up no space what so ever to store.
Second the recomendation to ship stuff instead of pay the airline to transport it, and remember, if the stuff in the box is not worth what it costs to ship it, then you may be better off leaving it and buying more stuff when you get to New York.