Unfortunately, hit and run accidents are common in drunk driving cases because some drivers want to avoid criminal charges and selfishly do not want to be held accountable for their actions. If that happens, do not try to chase down the other driver. All that will do is endanger others on the road.
Instead, take a picture of their vehicle if you can. If they drive away too fast, write down everything you remember about the car and as much of the license plate number as possible. Then, call the police. With your description they may be able to locate the driver, or check if a similar car was pulled over for drunk driving that same night. If the driver cannot be located, your drunk driving accident lawyer can still help you recover maximum compensation by filing an uninsured motorist claim with your own insurance company.