I would get your current landlord to write a reference for you. You can also get a policy on your dog so you can show potential landlords that you would have proper coverage if anything could happen. Do you know for a fact that your dog is an American Pit Bull Terrier. Its not a lie if you do not know for 100% fact what breed he is to call him a mutt. Then I would bring him with you when you go looking and let them meet your pup and decide then. Its not an easy battle but it CAN be done if you keep at it.