ABS light and Traction control light stay on. Vehicle brakes ok, but pedal feels "grindy" intermittently when braking

First check to see if all the wheel speed sensors are reading. If one fails, it will deactivate several of these systems.

How do you know brake pedal positioning sensor needs to be replaced ? Traction light on ? DTC'S - diagnostic trouble code's stored . That's why cruise won't engage . Taking your vehicle to a qualified repair shop an having it diagnosed is what you should do.
Cruise Control Inhibited
The engine control module (ECM) inhibits the cruise control operation when any of the following conditions exist:
• The ECM has not detected a brake pedal activation from the body control module (BCM).
• A Cruise Control System DTC has been set.
• The vehicle speed is less than 40.2 km/h (25 mph).
• The vehicle speed is too high.
• The vehicle is in PARK, REVERSE, NEUTRAL, or 1st gear.
• The engine RPM is too low.
• The engine RPM is too high.
• The system voltage is not between 9-16 volts.
• The Antilock Brake System (ABS)/Traction Control System (TCS) is active for more than 2 seconds.

Do the stop lamps work ?
DTC P0703 Brake Switch Circuit 2
The body control module (BCM) monitors the brake pedal position sensor. When the brake pedal is applied, the BCM detects a predetermined voltage signal. The BCM sends a GMLAN serial data message to the powertrain control module (PCM) indicating the status of the stop lamps.

DTC P1574 Stop Lamp Switch Circuit
The stop lamp switch signal circuit is a direct hardwire input to the throttle actuator control (TAC) module from the body control module (BCM). The TAC module monitors the stop lamp switch signal circuit in order to detect when the brake pedal has been applied. When the brake pedal is pressed, the BCM sends a GMLAN serial data message to the powertrain control module (PCM) indicating that the brake pedal has been applied. The BCM also sends a high voltage signal on the stop lamp switch signal circuit to the TAC module. The TAC module sends a universal asynchronous receiver transmitter (UART) serial data message to the PCM indicating that a brake pedal application has occurred. The PCM will compare the UART serial data message and the GMLAN serial data message in order to confirm that both serial data messages match.

The light being on means the system has detected a problem in one or more sensors. The vibration is the ABS modulating the brakes to prevent skidding because it has detected a wheel that is locked up. That wheel is most likely the one sensor that is giving you the error.

Grinding could be what's called "brake dust"... nothing can be done about that, take your vehicle back and ask them to clear the light.

That means there is a problem with the ABS , has DTC'S - diagnostic trouble codes in the EBCM - electronic brake control module . You need to have diagnostics run on it !
Traction Control/ Stabilitrak Off: Refer to ABS Description and Operation in Antilock Brake Systems
This vehicle is equipped with the BOSCH 5.3 antilock braking system.
The vehicle is equipped with the following braking systems:
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Antilock Brake System (ABS)

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Dynamic Rear Proportioning (DRP)

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Traction Control System (ETS) (w/NW7)

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Traction Control System (TCS) (w/NW9)

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Vehicle stability enhancement system (VSES)

One or more of the sensors in the wheels that are responsible for these functions is faulty. It is sending a signal that is mucking up the calculations that control the brakes and engine for traction control and anti skid. All of them need to be checked!

the traction control system works off of the abs system so there is a code in the abs module that needs to be pulled so you know were to start to check the system out.the service engine light is a different system all together and there is a code in the ecm that must be pulled so you know were to start to check to repair that system.you don`t say why you jumped to the conclusion that the gas cap set a code.the emergency brake light will come on if the e-brake is not all the way in the off position(something may be holding the lever up just enough to put the light on.the switch could be misadjusted and needs to e readjusted.

abs=anti-lock brake system/ tcs=traction control system ==sounds like you have a problem with ur abs possible bad sensor or a wiring connection problem when abs shuts down [trouble light] your brakes will work normal but if u stomp or brake hard the brakes will lock up skid and the tcs doesnt affect brakes but if u accelerate hard tires will spin lose traction check the wiring and connections to the sensors sometimes debris hits the wire and loosens the connection if looks ok and light is on have car scanned for the trouble code most auto stores such as auto zone/advanced auto offer this service for free the code will what and where the problem is so repair can be made

If brake pedal is chattering or pulsating it is most likely because you need either front or rear rotors, or both. They are rusted or out of round. If there is something wrong with either your abs or traction control,the abs or traction control light will come on & stay on. As soon as these lights come on, it automatically disables abs or traction control. If you would like to take out fuse that disables abs or traction control circuits, it will leave the lights on.You just wont have their functions. If by some strange condition your chattering goes away, then there is indeed something wrong with your abs or traction control.Fuses are labeled abs or traction either under hood or in your fuse panel in car.

NOPE IF ABS LIGHT IS ON AND YOUR BRAKES DONT FEEL RIGHT.HAVE IT TOWED.DONT DRIVE.