Turn signal comes on with brake lights

Stop Lamps
The brake pedal position sensor is used to sense the action of the driver application of the brake pedal. The body control module (BCM) provides a 5-volt reference voltage and low reference to the brake pedal position sensor. The position sensor provides an analog voltage signal back to the BCM that will increase as the brake pedal is applied. When the brake is applied, the BCM will then apply battery voltage to the left and right stop/turn signal lamp supply voltage circuits and the center high mounted stop lamp (CHMSL) supply voltage circuit illuminating the stop lamps. At the same time the BCM applies battery voltage through the signal circuit to the transmission control module (TCM) and the engine control module (ECM) indicating that the brakes are applied.
Ground for the stop lamps is supplied as follows:
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps
• G301 provides ground for the CHMSL
The stop lamps on this vehicle will not illuminate unless the ignition is in the accessory, run, or crank positions. When the ignition is in the OFF position the stop lamps will not illuminate when the brake pedal is applied.
Battery voltage from the LT TRN/SIG and RT TRN/SIG fuses located in the rear fuse block is also used by the body control module (BCM) for stop lamp operation.

Turn Signal Lamps
Battery voltage supplied to the body control module (BCM) from the fuses listed below is used for turn signal and hazard lamp operation.
• LT TRN/SIG fuse for the left turn signals
• CTSY/RF TRN fuse for the right front turn signal
• RR TRN/SIG fuse for the right rear turn signal
When the turn signal switch is place in either the LEFT or RIGHT position, ground from G300 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. At the same time the BCM sends a GMLAN serial data message to the instrument panel cluster (IPC) to cycle the appropriate turn signal indicator ON and OFF.
Ground for the turn signal lamps is supplied as follows:
• G104 provides ground for the left and right front turn signal lamps
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps

Do you have a factory scan tool (tech2) ? Your best bet , take it to the dealer .

check that all your bulbs are correct and are making contact. this commonly happens when a wrong bulb is installed in a socket. the other reason this happens is a bad ground, and one bulb uses another as a ground path. I would start at the rear left light and check that everything is good there, then move on to the front. 99 percent of these kinds of problems are at the lights themselves.

I would check the earths on the lights at the rear

Turn Signal Lamps
The LT T/SIG and RT T/SIG fuses located in the underhood fuse block supplies battery positive voltage to the body control module (BCM) for turn signal and hazard lamp operation. When the turn signal switch is place in either the LEFT or RIGHT position, ground from G202 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. The BCM also sends a message via GMLAN to the instrument panel cluster (IPC) to cycle the turn signal indicator ON and OFF depending on the position of the turn signal switch.
Ground for the turn signal lamps is supplied as follows:
• G101 provides ground for the left front turn signal lamp
• G100 provides ground for the right front turn signal lamp
• G302 provides ground for the left and right rear turn signal lamps

The LT T/SIG and RT T/SIG fuses located in the underhood fuse block are also used by the body control module (BCM) for stop lamp operation.
The brake pedal position sensor is used to sense the action of the driver application of the brake pedal. The body control module (BCM) provides both, a 5-volt reference voltage and low reference to the brake pedal position sensor. The position sensor provides an analog voltage signal back to the BCM that will increase as the brake pedal is applied. When the brake pedal is applied, the BCM will then apply battery voltage to the left and right stop lamp supply voltage circuits and the center high mounted stop lamp (CHMSL) supply voltage circuit illuminating the stop lamps. At the same time the BCM signals the transmission control module (TCM) and the engine control module (ECM) that the brakes are applied. Ground for all stop lamps is provided at G302.
The stop lamps on this vehicle will not illuminate unless the ignition is in the accessory, run, or crank positions. When the ignition is in the OFF position the stop lamps will not illuminate when the brake pedal is applied.

Did you check the fuse's. ? Wouldn't think it's a ground because right an left tail lamp Assembly's use same ground ! Did you check for B+ voltage at the bulb. with a volt meter ? Power is supplied by the BCM . Could be the tail lamp assembly

The bulb for the turn signal/brake light is a double filament bulb. It will come on with the lights on, but should light the second filament, making it brighter, with the brakes applied or turn signal on. It could just be a bad bulb, but one common problem on these years of truck is the rear light circuit board and requires the whole assembly to be replaced. Start by checking the bulb first though.
Hope this helps. Good luck!

Have fitted the proper bulbs some have a staggered pin on them only go in one way

Sound's like their is a problem with the right hand rear light cluster.The reason it only happen's going down a hill is you applying the brake. The only place the brake and indicator light's come together is the rear clusted. To check this apply the hand brake gently with the button held in instead of useing the foot brake. This may not be the fault but it is worth a try as it seems the most likely problem.

Sounds to me like a bad ground. When the current can't complete through a ground circit it feeds back through the elements ancd can cause all sorts of wierd syptoms.

The fast blink itself can be casued by a burnt out bulb I would check that first.

Then I would check the trailter harness if there is one and unpluch any harness that may be plugged into it. also check each bulb to see if it is seated well. you may be able to see if there is corrorsionaround the bulb terminals at this time is so yo may need to replace the light assembly. Also you could look for any spot where maybe someone in the past tried to tie into the harness to hook up additional lights etc. this may allo moisture in and corroded to the point the wire is "open"

sounds like a bad turn signal switch

if you have the circuit board type rear lights the circuits inside will brake and cross eachother that will cause your problem