I'm not sure about Long Island, but I did go visit a breeder up in Hudson, NY. Here's a picture of the beautiful puppy I met. His temperament was wonderful. I almost got him for myself, but the timing wasn't right.
Also, you can look at this website. It's a list of breeders who are recognized by the founders of the Australian Labradoodles- so you know they are reputable. Internationally Licensed Breeders
That's not to say that others aren't reputable, but this is a starting point. You can also do a google search. I think I found a breeder on Long Island, but I don't remember who they were. If you're looking for a breeder near you, then you can ask to visit the property. Most breeders won't let you see the puppies, but they should at the very least let you see where the puppies live, etc.
The best advice I can give you is to start talking to the breeder, those that you feel confident with instantly should be your top choices.
I just have to ask
Why just a Australian Labradoodle?
are you aware that most of them have spaniels and terriers infused in them? which adds more health issues and diminishes the non shedding aspect also?
also that the F1B Labradoodle is the most likly to be non shedding and allergy friendly of ANY? also since they havnet been line bred or inbred as some of the australian breeders do ,they will retain the hybrid vigor
From what I've seen, if you want an Austrailan Multigen you will need to spend $2000-2500. We could not spend that much so we purchased a F1B. My huband has allergies and F1Bs (F1 labradoodle x poodle) are more likely to be allergy friendly.
I read your other posts and from your bad experience I would get a puppy (as young as possible - 2-4 mos old) to make sure the puppy is socialized with your children as young as possible. In addition, their first fear stage occurs early (I think 2-3 mos) so if you are around for this period of the dog's life, you will know what fears the dog may have and why. I bet your first dog bit your child because of some fear.
I have met Multigens at the dog park and they have all seemed great - very calm and adorable. However if you cannot afford the high cost, I have found other labradoodles and goldendoodles to also be great.