There is not enough information given here for me to answer your question. But I am a dog lover, too! We currently have 3 dogs and a cat in the house.
First, we would have to know how much rent you can afford, and then we'll establish a price range that you are willing to look at. Second, we would have to know what areas you and your family are willing to live in.
Once we have the parameters of price and location set, then we can begin looking for houses to rent. The timing of your move may also be critical for you. If you are not ready to pay the rent for the next month, then it is too early to consider houses which are on the market now. But when I know when your current lease is up, then we will begin the search for a nice house at the appropriate time.
Your concern seems to be focused on a large dog you have. Most, but not all, landlords will consider a family pet once they know they have a good tenant who is submitting an application. So let's get YOU approved first, and then we can work on the landlord to get him to approve having a dog in residence. There will be a large deposit for a large dog, and some of it may be NON-refundable. Each landlord makes up their own rules. Small dogs, and even children, can cause as much damage to a property as a large dog can. It all depends on how they have been trained.
So my recommendation to a landlord is to go to the tenant's current residence and see how the dog is behaving there. You might even suggest that in your application to a partcular landlord that they come meet your dog at your current home. It might help if you offered a large deposit for the animal. (Money talks.) But most property managers have set rules, and many of them are prejudiced against large dogs.
If you will give me your parameters, I will gladly run a search for you. Then we can ask each landlord what their pet policy is.
ULTRA TReal Estate Services