You might want to look into a lab rescue group - I'm sure there is one or more out there that could provide temporary foster care and find a permanent new home for your dog. If I were you, and I'm not, I would avoid using the APL or some such organization. They are great places but there is no guarantee about the outcome of that situation. If you really want to make sure she gets into a good home, put the time into it by doing due diligence.
Another idea would be checking with friends and family (which I'm sure you've already done). Maybe someone you know wants your dog?
Finally, you might consider sending your dog to "doggie daycare" a day or two (or three) during the week. There are many around the area and that's a great way for your dog to get the exercise and attention they may not be able to get at home. It might only be a temporary situation - that is - once your kids get older, you may be able to give the dog more attention? Or, perhaps there is a kid in the neighborhood who could give the dog a good walk each day. I know that many people have home businesses that provide a range of services to pets.
Good luck with this difficult situation. I have two dogs and I understand that dogs are an important part of the family - I hope for the best outcome for you and your dog.