Let's clarify a couple of things on the tow dollies. There are two types one has a table on it that the front tires or back tires sits on and the table pivots in the center.
The other type the tires pivot and there is actually a tie rod going from one side to the other.
Now here's the part that you'll really like in both cases you lock the steering wheel of the car. That is to keep the car from pivoting into the fenders of the dolly.