Are there any hiking trails within driving distance? (I'm afraid I don't know Long Island very well!) What about the beach there? Are dogs allowed?
Dog parks are great for socialising dogs. But yes I suppose they could get a bit messy, especially if people don't 'pick up' after their dogs! And sometimes it can be tricky as you don't know which dogs are friendly and which are not.
Frequent bathing is obviously needed in the circumstances at the moment, but of course it dries the dog's skin and isn't normally a great idea. Could you ask your vet for a recommendation for a very mild shampoo which may be less drying?
If it's just plain mud or dust, then a wipe-down with wet or damp kitchen-roll paper towels will usually do the trick. I used rolls of the stuff with my dog, and her skin didn't suffer. But she was a short-haired mixed breed so it was easier.