Look into
Long Island Moving Company Men On The Move - I know lots of satisfied customers!
Oh, and I suggest you start packing non-essentials now. If you need sturdy boxes, many liquor stores leave their empty boxes outside for pick-up--they're free
The ones with inserts are great for glasses/cups, etc.
Start making a list of companies/providers who need to be contacted regarding your change of address; some are best notified well in advance, such as subscriptions to monthlies. Others will have to be notified regarding cancellation. Utilites such as water, electric and gas require reading of the meter for the final invoice.
BTW if you'll be leaving usable items behind and have no one in mind to give them to, consider posting them on freecycle.org (they have a Nassau, Western Suffolk and Eastern Suffolk Chapter). It's a national organization with chapters all over the country. People post items for pick-up ("offers") and some post "wanteds". The premise of the online organization is to keep usable items out of landfills. I highly recommend it! Have posted and given away many things, from furniture to plants, from books to clothes, etc. People offer appliances, tools - anything under the sun! And I've gotten some nice things I can use. :-)
90% of items I posted were picked up by someone who could use them!