how old is the dog and how big? Sometimes if you speak to your land lord and he gets to meet the dog and doesnt feel that your dog is a "liability" he will let you move in. Also, look into your state's "Pet Law." It is technically a loop hole and renders ANY lease useless, regardless of stated pet clauses. For example, I live in New York City, and our Pet Law is as follows:
The Administrative Code of New York City § 27-2009, commonly referred to as the "Pet Law," describes the rights of tenants to keep their pets under certain circumstances, despite lease provisions to the contrary. In its plainest reading, the Pet Law provides that once a pet lives in a multiple dwelling (a building with three or more residential units) for three or more months, open and notoriously (not hidden from the building's owners, their agents and on-site employees), then any no-pet clause in a lease is considered waived and unenforceable. A very helpful, in-depth explanation of the Pet Law can be found here (pdf).
My lease says im allowed to keep a cat or SMALL dog, however i realized i wanted a dog instead (building IS "dog" friendly) and I got him in Oct and he was only 12lbs so he WAS small when i got him. It's now Dec and he is 5 months and 34 lbs now, but it doesnt stop me from taking him on walks in and out of the building and the building management knows my boy and LOVES to play with him so he is DEFINITELY not hidden... its been 3 months, so legally i can keep him in my building. It also helps if when you meet your landlord that HE/SHE themselves are dog owners, they would be more likely to help you out. My land lord is a dog owner and said "i dont care what kind of pets you have as long as you dont have a tiger or lion in your apartment" LMFAO. Look into your laws, and find yourself some loop holes if you can't find an apartment that will take you...
I hope i was able to give you some ideas. Good luck, and keep us posted.