Brake/Gear Shift Fault

I've searched the forum but came up empty. I have a 2010 CrewMax 5.7L, 150k miles. When I apply the brake while driving, the reverse (R) and (D) indicator on the dash illuminate (R goes off when brake released). The backup camera also intermittently flashes in rearview mirror. I've disconnected the camera, still an issue. Vehicle is throwing a P0705 code. All lights are working. Reverse lights are not coming on when brake applied, but do when in reverse. I've tried clearing the code, but while the check engine light goes out the traction control light does not. Check engine light comes back on once we run again.

Car starts and runs fine. Any ideas??

Thanks in advance!
Don

 

Maybe the reverse lockout switch? Not the same thing but my old ram had a problem with the reverse lights not coming on. Ended up being the reverse lockout switch. I think that’s what it’s called

 

Thanks. Not sure if this is the same since brake lights are working. It seems like the brake wiring is cross-talking to the reverse, except that the reverse lights don’t come on when I hit the brakes.