Do you have a vacuum tester? If not, add this to the list for tomorrow.
Vacuum tester Autozone
Vacuum tester Walmart
Vacuum connectors
I would also get a foot of hose of 1/8", 1/4", 5/32" Inside diameter
These are almost a necessity and are fairly cheap. When you test vacuum you are not only testing the amount of vacuum, but if the needle fluctuates or not. Steady needle shows the valves are seating properly and a fluctuating needle can indicate a valve not fully closing, among other things.
So many variables to look at but always good to start with the basics and go from there.
Does it always stall when in
gear?
Does it
always stumble when you step on the gas?
Could the ignition switch cause these symptoms, mainly stalling?
What about the accel pedal or would this throw a code?
Why is it stalling? Loss of fuel, Loss of air, loss of spark, loss of compression, torque converter locking up.
Compression could be from a valve hanging open but I would suspect a hard stumble then it would vibrate.
Loss of air.....not sure how.
Loss of fuel or way too much. Injectors obviously don't show leakage or at least all the time.
Time to pull both the battery terminals, (Neg first)
Check all your ground connections, and even the positive terminals. Check the strap going to the transmission, I believe it has one. Take some sandpaper and clean both sides of the terminals you pull and the metal it contacts.
Anywhere you see a black wire attached to the metal around the engine bay, remove it and clean it.
Spark could be from a crank sensor, perhaps not having the CASE relearn....dunno. I don't know where the sensor is but I would find it and spray some electronics cleaner on the terminals.....allow to dry, dries fairly quickly.
I would pull the coil leads and clean all the terminals with the spray.
I would pull the TB harness and clean the terminals.
This is alot to do, trying to help diagnose this blindly this is exactly what I would do if it were mine, would cover alot of bases. Most of this you don't need to do, but I would do it anyway.
After pulling all my hair out trying to figure out why a TB would not readapt on our previous jetta, I have alot more respect for clean terminals.
Dealer couldn't find it, kept telling me I needed a new TB but I kindly declined. I pulled the battery and cleaned all the grounds under the battery tray (4). tried another TB, no dice. A buddy of mine used to be a VW tech and attached a small 16 ga wire from the ground terminal to the chassis totaling 5 now and it worked. Those electronics were extremely sensitive to current flow and voltage.
Edited to add: When you step on the gas, it takes alot more current (spark) to light the mixture, even more under higher loads. Could be an electrical issue and the coils aren't getting all the current they need, just a guess.
Fuel economy is dismal so could be the injectors.
Don't forget the 44K.