I work in the area, feel free to contact me. The one and only reason I could think that no one is responding or calling back is that your wants may not necessarily meet what is available in the area.
For instance, Center Moriches as of right now only has 10 listings for year round rent, and 1 for only seasonal..., but not all 11 may be readily available. Now out of this 11, if you have pets we only have approximately 2 that will allow pets. Now I have not made calls to check availability, but those 2 may already be gone?
With pets you are limited to just those 2 homes, now we must also weigh in how many people you have total. Depending on how many people will reside with you, will also cut that number down depending on the permit max allowance number the town has issued the home.
I am thinking that the agents you are calling should most respectfully respond to your calls, but these weighing in factors could be why they are forgetting to make contact, the reason being... there just might not be anything available.
If you would like some help send me an email email@example.com
Even with the limited amount of homes out there, I will try my best to match you up with something that works for your family.