Could I have hurt my health driving a car with no exhaust?

Your hearing MAYBE. But other than that no. If you think about it, race drivers, and their teams, spend hours nearly every day running, and driving cars with little to no exhaust system. What your tech was reffering to was you could have passed out from lack of o2 due to exhaust gasses acumilating in the cab. But you made it through no problem, there won't be any long lasting effects. What your feeling now is just anxiety.