I have a 2003 Dodge Ram 1500 Quad Cab. I was constantly getting burned out brake light bulbs on the rear left and right (mostly right) brake lights and the annoying intermittent chime even after replacing the bulb. I replaced both rear socket assemblies and it made great improvement. However, when I have the headlights on and activate the right turn signal, I get the bulb out chime (without the bulb out caution light) and the turn signal light flashes a bit quicker, but not much. When the turn signal goes off, so does the chime. Thoughts?

 

Obviously you have an issue with the right turn signal. Check the most common....bad grounds sockets bulbs and wiring. Tough to diagnose electrical thru websites. I had the annoying symptom and finally traced it to a burned contact on one of the bulbs. I rectified it by cleaning the socket contacts real well and replacing the bulb for a third time.