I must admit when I first saw the title of the post my first instinct response was, “Change the Cam Sensor”. That is typically 80 percent of the problem. Since you have changed it many times I would have to say it maybe something else. Do you know if there are any engine codes present? You may try to replace the pigtail harness that plugs into the cam sensor. It may have a short in it. Also there should be a rubber seal on the harness that keeps water from getting into the connector.
To me it sounds like there is a wire short. It maybe a wire short from the key switch to the relays or a loose ignition, or ecm fuse. The ecm fuse is usually a 40 amp fuse located on top of the batteries.
When you changed the cam sensor did the problem go away for awhile and then come back? I have also known those cam sensors to go bad often even after replacing them. Also when you installed the cam sensor make sure to follow the instructions about how many metal shims to use. It does make a difference. Also, I have had them give me the cam sensor for a different engine and they look the same and plug in the same. So make sure they are giving you the correct sensor.
Keep me posted.