Sorry, my original post wasn't as clear as it could have been. I'm using the DF supplied mirrors that have the running light and turn signal in them and both work correctly regardless of whether the headlights are on or off. No issues with the mirrors.My side markers do not stay on bright, they are dimmer as running lights and then hyper-flash brightly. Since you have an issue at both the side markers and rear lights staying on bright as DRL's I would start with what's common between them. But GTStory is right, we can only give suggestions, the hard work will be with a volt meter. Unfortunately that might involve looking through AllData which is a little funky to find stuff at first.
I just looked at my mirror/side markers. There is two light sections. The outer most section of the turn signal/mirror is the DRL, the inner most section is the signal flasher part of the unit
My tail lights are the problem. With the headlights on, the tail lights are continuously on bright (like the brakes are on) and they don't blink when the turn signal is activated. With the headlights off, the turn signals and brake lights work correctly.
While my frame was at the powder coater I opened the tail lights to determine which wires were for which function. I confirmed that they have a dim light (running light), a bright light (stop/turn) and reverse, and all three functions worked correctly on my bench.
I connected the running light wire in the tail light to the brown wire in the harness, and the stop turn wire in the tail light to the green/yellow wires in the harness. I just find it odd that the stop/turns work when the headlights are off, but not when they're on, and the running lights are bright instead of dim when the headlights are on. And the turn signals work when the headlights are off, but not when the headlights are on.