Sometimes since they need care right away I will offer them the option of 2 weeks of care while we sort everything out. That gives both of us an out and as someone noted that out of control child can be a dream child in the right environment.
You talking to the current caregiver might be the best way to go - sounds like you need to prove to her that you are ok to take over one of her charges and the parents are relying on her for guidance too.
One thing that is coming out of this is that the family is taking advice from the caregiver. Ideal if they develop the same level of rapport with you. Bad of course if they can't make the break from the old caregiver.