If all the running lights (front & back) are not working I'd start by checking the fuses then onto the headlight switch
I would check the running lights for one with a short. I had the same sort of thing with my trailer and I spent way to much time troubleshooting the brake and signal wires. Turns out a running light bulb connector was grounding out on the trailer and messing it all up.
The whole turn signal arm unit needs to be replaced
to differentiate it from a brake light or running lights. Its the only one that BLINKS at a regular rate when looking at it from the back.
The only thing that will work with key OFF is the brake lights,head lights and emergency flashers. Dome light will also work.
the turn signal switch is probably bad
Fuse
If you have the same bulb for brake and signal, brake light power needs to be interrupted by the signal lights, The brake circuit has to go through the signal light switch so the problem could be there.
Be sure to keep the rear tail lights, brake lights and signal lights in good working order. To many times you see vehicles with lights missing.
On most cars, the brake light and turn signal are the same element in the bulb with two elements in it. The other one is the tail lights. The brake light will often burn out sooner because it is used more and the on off is harder on the element. If you have the front amber running lights, it is almost certainly the bulbs.
Since most brake/turn use the same filament in the bulb(1157) then if the turn signal switch is bad or offset then it will effect the brake lights.
Start with checking the bulbs, then the fuses. In most vehicles the brake lights and siganal lights and running lights are separate circuits. If the fuses and bulbs check okay check the wiring feeding the rear of vehicle.
fuse, bulbs