The rad. Fan runs all the time, and continues to run even with the ignition off

Most auto fans are designed to run until the temperature gets down to a certain level, even with the ignition off. But it shouldn't be running long after the engine is cooled. Maybe there is a bad sensor. It will of course, run the battery down if it runs continuously.
Check the owners manual. If it is definitely a problem, find a good mechanic or radiator shop to look at it.