I have a '85 500SEC. It had/has a stalling problem also.
What I have found out is: The fuel pump wasn't getting any voltage. That is the fuel pump relay.
I popped the cover off the relay and you could visibly see the contacts on the solenoid close.
At last, out of frustration, I was handling the relay and I noticed that the support for the solenoid was loose from the circuit board!! There are two solenoids in there. The small one is the over-voltage. The large one is the fuel pump.
I took my soldering iron, and re-soldered the back of the support. Apparently, the support is the ground. It was "cold-soldered" at the factory. It also has a bit of corrosion on it. That is because, as the current flows through it, it developes disimilar metal corrosion.
Anyway, just re-solder it and plug her in.
I had bought a used one, in the beginning, and this one had the exact same problem. I fixed both and carry both. Good luck, John F. '81 240D; '85 500SEC.