Well at 13 a snotty attitude is pretty normal. It is how the parents deal with it that will determine whether it will get better or whether it will spiral out of control. Your husband has to stop letting her make him feel guilty and babying her, or it will spiral out of control.
First and foremost, you and your husband need to set some ground rules. Do what you are told, when you are told and not 15 minutes later. Do it with a kind attitude. Treat other family members with respect. No name calling. Etc... Once rules are agreed on and explained to all the kids (don't single her out

) there have to be consequences when she ignores those, which she will, at first.
My first thought was the cell phone. You said she was sitting in the back seat texting? You might consider that as a vehicle for discipline. If she wants to have cell phone priveleges she better keep her attitude in line. When she starts ignoring parental requests and making snyde remarks, you or her dad can say, "Hand over the phone. You may have it back again when you are ready to be respectful."
Of course she will have a temper tantrum and when her mother finds out she will insist that she must have her cell phone in case of emergency. Then your husband can explain that while she is with him, he will handle any emergencies, and it is his house and these will now be the rules and refuse to discuss it further. He might also need to have a talk with his ex pointing out that he is doing his best to be a good father and he really is helping out with a lot of the hands on child care. Doesn't make him a saint or anything, but it is a lot more help than a lot of divorced moms get. So any way that she can be supportive of the rules in his home would be appreciated.
It is a tough situation for you because you are not her biological mom, but her behavior will affect your children regardless of DNA. I hope that you can take her in hand, before she has the 3 younger ones thinking that is the way to act when you don't get your way. Good luck.