I always spray wd40 into my 7-way trailer connector before hooking up. Then wiggle the plug around for a few seconds then lock in place.
Usually if some of the trailer lights are working and others not it turns out to be a dirty connection.
I think I read in my 2010 Ford book that all of the lines feeding the trailer connector are separately fused. I never confirmed that.
The RELAY AND FUSE you was referring to is for sending 12VDC Charge voltage back to the trailer. Should not have anything to do with the regular lights.
Kinda important that you have a good trailer connection in case your trailer is equipped with electric brakes. It is very important for those to be working properly. It is also a DOT LAW that they are working as well...
I picked up one of these 7-way trailer connector adapter plug to test my connection. I got mine from AMAZON or maybe it was ETRAILER. I have seen them in auto parts stores.
Very useful to have in your RV TOOL BOX. If all the small leds work properly when you activate the trailer lights then the problem you are having with your trailer lights is on the trailer side of things...