Can I drive my car without the Catalytic Converters?

My car's Catalytic Converters were removed before I had gotten it, but the exhaust pipe was welded back to straight pipe it. Well, when the last driver that trailered it was dropping the car off the winch hook to the trailer had gotten hooked up under the car without him knowing and as he took off it ripped the exhaust down. The exhaust was still attached to the car, but it was never fixed from whatever damage that may have caused, but recently I was driving it and the exhaust pretty much fell out of my car from underneath and the only way at the time to fix it was to cut the exhaust pipe from the front and leaving it short. My husband told me I can no longer drive the car until the exhaust pipe is fixed because driving with a short exhaust pipe could blow the motor on some cars, but he couldn't remember which ones. Can I continue to drive my car without getting it fixed?