State laws as well as insurance companies' guidelines can differ regarding if insurance transfers from a car you had insurance on and replaced with a different car. Some insurance companies extend the old vehicle's insurance onto the new car for a few days so you have time to come in and change the information, others may not allow this.
This transfer of coverage will be the same type of coverage you had on your trade in vehicle. So if the car you took in only had liability and the car you drove away from the dealership requires physical damage coverage on it, than you will not be covered for that "full coverage" until you add it onto the vehicle.
If you do not notify the company of the change of vehicle within the given time, allowed either by state law or the insurance company, then your new car will not have coverage on it.
To find out if your insurance company was insuring this new vehicle at the time you were pulled over you should contact them. If the car was insured under your previous car's policy, get documentation stating this and see if the Wyoming court dealing with your ticket will accept this as proof of insurance and dismiss your ticket.
For information on what Utah's insurance laws are and if the state requires a change over or grace period when replacing one car with another on your insurance policy, contact the Utah Insurance Department.