Iowa has this issue, and it's actually a little worse than that there, as there is no way to "quit-claim" a car.
This came up in my divorce, actually. She was getting a car that had both our names on it. Iowa law says that the owner of a vehicle is liable for whatever happens with it (which, of course, is what liability insurance addresses). By the time the divorce was actually finalized, I had been living in another state for almost a year. I was attempting to follow the decree and sign over title to the car, except that Iowa has no mechanism for doing so. The new owner has to complete his or her part of the process, and let's just say that cooperation had been an issue near the end.
Now, she should have been able just to take the decree and the title to the county administration office and get the title transferred, since the decree said it could be used in place of a signature in order to carry out its requirements (a decree is a court order, of course, so it has a judge's signature on it). But, I just have to hope she actually did this.
When I tried to get car insurance for myself in my new state of residence, I was wondering why the rates were so much higher than I was paying in Iowa. I found out when I got a call from the insurer I eventually used to go through my driving record, etc. It turns out that she had two accidents in that car between when I filed for divorce and when the divorce was finalized that she never told me about, which is a big problem since legally, I was liable. She still had insurance, apparently, which is good, but the accidents were initially making my insurance go up since they were recorded as being in my vehicle. The insurance agent had that corrected, and my rate went down.