During the off season, from now until the middle of May, you can get your own room with a shared bath for $700/month or less. From the middle of May until early September, it will cost substantially more.
Check out Craigslist, Dan's Papers, the Independent, and the East Hampton Star. There are many rooms available; and plenty of houses are dog-friendly. There is a great 40 acre dog park off Three Mile Harbor Hog Creek Highway in the Springs section of East Hampton.