You should be able to get your own plan. Because you are in New Jersey, all people in the same household must be on the policy unless you can show proof the other people you live with are insured elsewhere.
If your daughter was the listed driver and the accident was logged correctly, most companies won't charge that against you. If you are looking to switch insurers and her accident comes up on your claims history, I would obtain proof or confirmation from that company that the claim wasn't yours so it can be cleared.
I would strongly advise not letting your nanny drive your kids without insurance. If she were to get in an accident, no coverage would be in place and she would have to foot the bill for damages to all vehicles and potential injuries for your children.