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Which code was set? Need to know for daig help. As far as why the trans shifts hard, it is because the Transmission Control Module commands max line pressure and turns off the shift adapts when it loses the signal from the ISS
How long did this go on? What were you doing when it happened?
Check your fuses first. There is an "EPS" 60A fuse in the underhood electrical box, and an "EPS/STRWHL CNTRL" fuse 2A on the BCM fuse box (in position #19) inside the vehicle.
Does it crank over when you try to start it? Do all the lights come on with key on? Need more info.
How long does it take the battery to die? It sounds like you have a parasitic draw on the battery. Charge the battery first. Then disconnect the negative battery cable from the battery and install a draw tester inline on the negative battery cable. One end of the tester connects to the Negative battery post, and the other end connects to the negative battery cable. Close the tester so it is part of the battery cable first. Drive the vehicle and use EVERY accessory during the drive (radio, wipers, headlamps, flashers, defroster, A/C, etc.). Park the vehicle and open the hood, then hook up a DVOM with the AMP setting (every meter is different, but use the setting between that says "A" not "mA") selected. Wait at least 3-5 minutes before doing the next step so that all the accessories and modules in the car have a chance to go to sleep (and make sure all the door are closed so the dome light doesn't give you a false reading). Then open up the Draw Tester and see what the meter reads. If the reading is high, start pulling fuses one at a time (you may have to have someone in the vehicle to do this as I don't remember if you can reach the fuse box from outside without opening the door) until the reading drops within the acceptable range. You then know what circuit to check for loads causing the draw. You will have to check on the battery itself or get specs from a service book or the battery manufacturer's website to know how much of a mAh (mili-amp hour) draw the battery is designed for. If you are unsure about how to do any of this, seek help from a qualified professional in your area.
Diagnosing the rough idle when warm from an armchair is hard. I can tell you that you have (at least) 2 seperate problems. Rough idling when hot is engine related, and grinding/shaking in the body during braking are not connected. Where does it shake when you apply the brakes? Front end wobble or like the whole vehicle vibrates? To figure out if it is front or rear brakes, apply the brake pedal at about 45 mph and see if you notice the steering wheel shake/move side to side. If it is front end wobble, you should have the rotors checked for runout, the wheel bearings repacked and the steering and front suspension checked for slop/play. If the wheel does not shake and or move, but you feel it in the "seat of your pants", then try applying the parking brake (carefully) at 40-45 mph without touching the brake pedal. To do this, keep the park brake release handle pulled out and push reasonably hard on the park brake pedal with your foot to apply them. If it is in the rear brakes, the vibration will be there without the front brakes applied. If it's rear, have the drums checked for out-of-round problems, and the rear shoes and hardware checked for wear and or loose/missing hardware. May also have a rear differential issue.
Is the stalling usually worse or mostly noticeable when the vehicle is warmed up and after coming off the highway/freeway from a long cruise, or at lower engine RPM under a load (like turning the steering wheel) or does temperature, engine load or engine speed make a difference at all? First I would say check the distributor components. Pull the distributor cap and check the condition of the cap and rotor contacts. Also look for the dreaded "green fuzz" corrosion commonly seen. If there is green fuzz on the wiring at the control module, or where the module mounts to the housing, disconnect, remove and clean all that away and reinstall it with some dielectric grease on the mounting side and the connectors to the wires. If the green fuzz is around the reluctor wheel (shaft center) or the windings around it, replace the distributor. If there is no corrosion, and no burned cap and rotor contacs, then take a small metal (can't be aluminum or anything that won't stick to a magnet) screwdriver and touch it to the reluctor wheel on the shaft. Try and then pull it away from the reluctor wheel. You should feel some resistance (pull back) from the magnet. If there is not magnetic pull to the screwdriver, you have a weak magnetic reluctor wheel and need to replace the distributor. Also check that the timing connector (usually a single wire black connector (with a Tan wire w/ black stripe) located either close to the distributor on the wiring harness, or under the black plastic cover on the firewall that houses a power distribution center, or in the vehicle on the passenger side in the wiring harness just below the heater box. Make sure it is plugged in securely.
While it is possible that you have a module or input problem, I need some more info. Is the camera on the Rear View Mirror, or do you have the Navagation Radio with the camera display there? If it is on the mirror, then lift the mirror up and check that the electrical connector is seated all the way (we usually reccommend unplugging it first and checking for loose terminals, then plugging it back in firmly). If it is on the Nav Radio, there is a software programming update for this concern. Have it programmed before looking any further. Next: Are there any aftermarket electronics or modifications installed? There have been some real issues with some of them. Let me know.
Should be on the A/C Discharge hose/line from the compressor to the Condenser
Pressure test the cooling system for leaks. If you don't see a leak at about 16-20 PSI, check for one at lower pressure too (about 7-11 PSI). The 5.3L V-8's have had a common water pump leak occurring. Just something to check.