1994 Nissan Altima Gasket or One-Time Issue

My radiator went bad and my car overheated. I had it towed to a national car repair chain. They replaced the radiator. I drove the car the next day and after about 30 mins, I had very little ability to accelerate out of 1st and 2nd gear. White smoke began coming out of the exhaust. As I was driving back to the repair shop, the car overheated and all the coolant was gone. I filled the radiator with water and it ran fine the remainder of the trip back to the repair shop. I'm awaiting their inspection of the problem now, but I'm thinking coolant may have gotten under the gasket heads when the car overheated causing my issue. What do think?