what a hard situation to be in, maybe you could talk to the RSPCA, they sometimes have people who temporarily home animals, in the homes rather than in the RSPCA place in a horrid cage. put an add in the paper, be honest, you never know if you dont ask.
You have an opportunity to give your self and your children a great new start and improve your lives, it would be silly not to take it and then in a few months have the opportunity go and the dog pass away.
i know she is a loved family member, but there are other members of the family to think about too. maybe others wouldnt agree, but thats my thoughts anyway.