You might want to first verify that power for the running/tail lights is making it to the 7 pin connector at the back of the tow vehicle.
Then check to see if it's getting from the 7 pin to the other end of the pigtail. On Sunlines of that era, the pigtail terminated under the trailer just aft of the A-frame
It split there with 12vdc hot going up into the trailer to the hidden 30 amp master fuse. The brake wire went from there to the brakes.
Brown, green, and yellow continued under the trailer all the way to the rear. Green an yellow are the brake and turn signal lines. Brown is the tail/running lights.
It should be a simple matter of tracing and testing the brown wire using a multi-meter or even just a 12vdc test light.
'12 F250 4x4 Super Duty PowerStroke 6.7 diesel
2011 to present: '11 Cougar 326MKS
1999 to 2011: '99 Sunline T-2453
Amateur Radio kd2iat monitoring 146.52