If you are committed to having a dog in your family, then please don't give up your search just because you've been denied by a couple of rescues. Based on the amount of time both of you are away from the home, I would highly recommended an adult dog, and forget about the idea of a puppy. Puppies are very demanding of your time and attention.
Although your neighbor's offer is very kind, don't rely on it, who's to say after a couple months they might get tired of helping. I think it would be fine to let them come over and play if they want to, but don't take on a responsibility thinking you will always have help.
Please don't give up going through a rescue. It's just a matter of finding the best dog for your particular situation.
Do some research on specific breeds you both agree on as far as looks and then look at the typical temperment/characteristics of the breeds and see if there is one or a couple that you both agree on. Read your newspaper, look at the classified section of petfinder.com if you want to avoid a rescue. There are tons of adult dogs out there who need a good home. Determine what problems you are both willing to take on and what problems you want to avoid completely, such as barking, will they be okay with children, if you don't have children around that might not make a difference.
The best part about finding a rescue who will work with you is that most rescue dogs are coming from foster homes and they should be able to tell you a LOT about the specific dog you think you might like. If you get one from a pet store (and I beg you not to even consider that option!) or from a personal ad, or the pound, you may end up taking home a very wonderful addition to your family, or a wild card with a lot of baggage and behavioral problems.
Once you get a more solid idea of what you are looking for in a dog, then research rescue groups. Some will be local, but there are also rescues that work nationwide and will transport an animal to you (with or without additional cost to you). I would recommend contacting a few rescues and just explaining your situation and what your looking for. They may have a dog that they feel would be best suited for you or if not, they may at least keep you in mind for the future.
The rescues that I know of and have had direct contact with lately are getting new dogs in almost every day, so you may find the "perfect" dog for you very quickly.
Another plus about going through a rescue is that most will make sure the dog has up-to-date vaccinations and be spayed/neutered prior to adoption.
I wish you the best of luck!!