[67C10Step]

Hate to say it, but almost every time I've seen running lights go out when the brakes are applied it has been a grounding issue. Try running a designated ground wire from the light that is going out to a known ground. If that clears the issue you know it's a ground problem.

Other than ground, the two circuits should not have much to do with one another. The only other place might be the directional signal switch.

Someone else will be along with probably the exact fix.