If you use the car "dolly" the front of the car is off the road and on the dolly. If loaded properly the front wheels don't spin and therefore no danger of harming the auto trans. What is your tow vehicle? You will use more fuel towing the car and you might need to use magnetic brake lights attached to the car for safety.
If for some reason someone can't stop and hits you from behind, they will hit your car and likely push it off the dolly and into the truck. A good hit will damage the rear, and likely the front of the car. If you don't have full insurance coverage this could be a financial disaster. Things to contemplate as you look at your options.
Towing the car behind a U-Haul type truck on a dolly will increase your fuel costs and risks. You are driving an unfamiliar truck with about an extra 15' extended onto the back. You'll need to watch it on corners, in parking lots, and backing up will be all kinds of fun.