Cars STILL used freon... just a different kind.
Cars built pre-1996 used R12. This is the freon that was banned because it was found to destroy ozone. As of January 1, 1996, the auto industry stopped using it (so 1996-model cars that were already built before that date may have still used it, depending on the company's policies... those built after that date should not).
Cars built in 1996 or later were required to use 134A, an ozone friendly freon. So if your car is a 1997 model, you should be in the clear.