Install a 75 micron fuel filter into the fuel line coming from tank. Make sure fuel line to carb (from on/off valve ) is not obstructed. There should be a steady continuous stream.
Loosen or remove fuel cap (as a test) to see if that helps. If runs ok without fuel cap, replace fuel cap,
If fuel cap doesn't help, turn off fuel and remove fuel bowl, turn on fuel. Again there should be a steady flow of fuel. If not, remove float and inlet needle and clear obstruction. Reinstall needle, float and fuel bowl. If still wont run properly, order a chinese replacement carburetor ($15 US delivered) and replace carb. Then drain fuel from tank and fuel bowl end of every season.
Suggest you first check the seat switch. Most deere riders you can start/run engine with brake set while not in seat. But operator must be in seat when brake is released. Your in the seat and it dies, so seat switch failure.
But here's instructions for brake switch.
BRAKE SWITCH TEST AND
To make sure the brake switch has continuity when the
brake pedal is depressed.
1. Turn key switch off.
2. Depress brake pedal and engage park brake lock.
NOTE: Brake switch is located on left side of frame
3. Remove footrest.
4. Disconnect brake switch connector.
1. Check continuity across terminals (A). Check
continuity across terminals (B).
NOTE: Be sure actuator (C) contacts switch plunger
(plunger depressed) and not switch body.
2. Release park brake pedal.
3. Check continuity across terminals. Be sure actuator
does not contact switch plunger (plunger
Sounds like lack of fuel. Try backing the air fuel mixture screw back 2 turns and see if it helps. Once it starts, see if you can keep it running by spraying carb cleaner into the carb. If you can keep it running long enough, place the palm of your hand over the carb inlet to choke the carb causing a sudden suction in the carb. Some times this will dislodge foreign material and help it run.
The videos are not the exact model but are an excellent primer.
Check that belts and friction wheel/disc are in good condition. Look for loose linkage in the direction/speed select. There are kotter pins (some models) that may fall out of the linkage connections and cause issues with direction and speed control.
Remove transmission cover for easy access, see 2nd video.
Select a forward speed and note location of friction WHEEL relative to friction DISC. Better yet, mark the DISC with a sharpie where the WHEEL meets the DISC. Now select reverse/speed and note the friction WHEEL has moved to a point on the opposite side of the center of the friction DISC.
Moving the wheel to the opposing side of the disc effects change in direction. Moving the wheel farther from the center of the disc increases directional speed. A simple, yet effective, and relatively dependable mechanism.
So to increase reverse speed, you need to adjust the linkage so when you select a reverse/speed the friction WHEEL is at a point further from the center of the friction DISC.
I havent done one of these in awhile. However, study the following link which is the diagram for the dirive section of your model.
Note part #s tagged 40 and 41. I'm reasonably sure this is where the adjustment is made. Before you make changes, follow the shift lever mechanism from knob to friction wheel and look for missing or loose bolts and pins. Also, video 2 showed how to determine a worn friction wheel and disc that needs replaced. Snap several before and/after pics from different angles for your records.
Further; This adjustment will decrease the forward high speed, but that's likely not a concern.
Check ignition kill shunt. It is a black/white wire the runs from the "M" pole of the back of the Start/Run switch to the ignition module (underneath engine cowling).
With switch in OFF position, the M pole should have continuity with ground (Bat -). With switch in ON position, it will have a high impedance (few k ohms) or open relative to ground. If switch and wiring check OK, likely ignition module has failed or damaged (mice).
If belt is ok, then likely the rear wheel dogs have seized. Review the following PartsTree diagram, what they call ratchet key item 3.
When the mower is off, you would normally hear the dogs click as you pushed the mower. These get dirty from debris and then rust up and seize (no longer click). Once seized, the transmission drive shaft just spins inside the drive gear.
Remove bolt wheel and wheel to access and clean out all the dirt and debris. Inspect inside side of rear wheel and verify wheel gear is in good condition. If all looks good then onto the drive gear.
You could try penetrating oil, but if dogs are seized you'll likely need to remove the gear... the key is spring loaded so take not to loose the key and spring. They're small pieces and once they launch they may be lost forever.
I don't remember if the "cover" has to be removed. If it does, there are a lot of small parts, be careful not to loose and keep in correct order. Consult diagram frequently.
I have a yard works trimmer and it does the same thing. To keep it running i have to put the choke at half way on. I took it back to the store where I bought it and they made some phone calls. The outcome is they said the controls are set up for the country where they make it. this country is very high and the atmospheric pressure is a lot different than ours. His advice to me was if I am happy with it working that way just keep it. So far it starts and runs good but it has to be choked all the time half way.
The parts diagrams online show some little parts in the drive train, thrust washers and idler/drivers. Are all those parts accounted for and in place? Is the polarity on the motor turning it the proper direction? Hope you can run it with a cover open to see what is happening or not happening. Can you turn the paddles by hand and the motor turns with it?