Take another look at those blue claws on the beach and you'll notice that they are not dead crabs, but actually just empy shells. Crabs are like snakes; their outer skeleton cannot grow. They grow a new shell underneath and when they are ready, they shed their old shell just like a snake sheds its old skin. This is called "molting".
For the few days following the molt, the crabs are "soft shell".
This is prime growth time for blue claws to molt and it's not uncommon to see hundreds of their shells float up on the beach. Every so often, a good crab molt even fools marine biologists into thinking that there was a major crab die-off. Here is a good article that tells how even scientists are sometimes fooled by a crab molt into thinking that crabs were dying...
The last few years there have been some great blue claw crab population blooms in your area. Each time there is a major molt, people flip out and think that the pesticide spraying is killing the crab population. The truth is, a good molt is a sign of growth, not death.
Keep in mind, clams are their primary food source. Clams love BRACKISH water with low salinity and lots of floating food particles in the water (what you and I would consider "dirty water"). When the clam population increases, so does the blue claw crabs.
The second major food supply of the blue claws is the young, old sick & weak. Crabs are "opportunist" feeders. They'll play the roll of the scavenger, the hunter, the stalker... whatever works at any given moment. If the water quality is poor or the oxygen levels drop and fish begin to weaken and/or die, the blue claws will benefit.
Is the sewer treatment plant still pumping waste into the Patchogue river? If not, when did it stop? Either way, that might explain the smell in the Patchogue area. But what about Sayville?