Barking dogs can potentially be an issue, but you describe a situation in which you rented with explicit knowledge that there would be barking dogs. Did you see the dogs before you rented? Are these dogs particularly loud, as compared to average dogs (or average dogs of this breed)?
Assuming you didn't agree to the contrary in a written rental agreement, for a month-to-month tenancy you would normally be required to give 30 days notice of your intention to vacate. If rent was paid through May 9, and a thirty-day notice period would have ended on May 21, that would mean that your landlord should at most be charging for 13 days rent and refunding the balance of the deposit.

