Insurance on a Car I Don’t Own?

(USA-Kentucky) My wife and I purchased auto insurance on two cars a few years ago. I didn’t notice at the time, but the policy had a clause where they wouldn’t increase the rate unless we made changes to the policy.

This year, I sold one of the cars (don’t plan to purchase another for a while) so I notified the insurance company. The employee told me that I would be receiving a bill for $400 because my change to the policy would trigger an update, requiring a $ increase. I was surprised, because I thought my rate would drop, due to only insuring one car.

I then asked the employee if it would be OK to just keep the old (sold) car on the policy, so that I could continue to pay the old rate. She was OK with that.

So my question is: Am I taking on any unnecessary liability here? If the new owner has an accident with the car, will his new insurance company know about the coverage from my insurance company and attempt redress, and would that put me on the hook for anything?