I am looking at a diagram for a 2003 superduty. It does have a fuse 38 in the underdash fuse box. I believe this setup is the one that has the running lights relay built into the fuse box and is not serviceable. You should have a brown/white wire coming out of the fuse box and going to the trailer plug. There is a solid brown wire feeding the relay inside the fuse box. When you turn the running lights on, the brown wire should have voltage. This triggers the relay and then the relay hooks fuse #38 to the brown/white wire, and sends the #38 voltage back to the trailer plug.

I would get a meter or testlight, find the brown/white wire at the fuse box, and see if you have power with the running lights on.