If selling a car, do you save paying tax trading in your old car to the dealer?

That seems to be correct in most states. Here is a citation.

“Trading-in may also offer a tax advantage if you are buying a vehicle at the same time. In most states, according to dealers and regional authorities contacted by MSN Autos, when your car is taken in trade you only pay sales tax on the difference in cost between its trade-in value and the price of the new car. For example, if a dealer gives you $10,000 on your trade-in and the purchase price of the car you are buying is $25,000, you’ll only be required to pay sales tax on the $15,000 difference between the two amounts. In states with a high sales tax, this benefit can help narrow the difference between trade-in value and private party price.”

By the way, that article also talks about other benefits of the trade in that may offset the lower price that you get. It absolves you of liability to a buyer of your old vehicle.