No brake with truck running.

Having an issue with my 1994 Sierra K1500. The brake lights all work with the truck off. If I start the truck the lights in the bed stop working. Third brake light still works. I know the bulbs are good because the hazards and turn signals work. If the truck is running with the hazards on and you step on the brake pedal the hazards keep on flashing. Shouldn't they stop flashing? I replaced the multi function switch because of other reasons but interestingly with the multi function switch disconnected the bed brake lamps do not work no matter if the truck is running or not. Plug in the multi function switch and they work, but again only with the truck off. Checked ground on driver side harness under bed. It's good. I even cleaned the metal just in case. I have not checked the brake light relay yet only because the third brake light works like it should. Is that on a separate relay? I know it is on a separate circuit. Also oddly with the truck off, when you step on the brake pedal the electronic climate control powers up. Release the brake pedal and it shuts off.