My 560 SEL has a very strange problem when it is restarted while still warm. (I know--vapor lock immediately comes to mind, but...). This had been an intermittent problem that only occurred in spring and fall, but has recently degenerated, and is a problem almost every restart, now. The symptoms:
When the car is initially started cold, everything is perfect. When it is shut down for a short time, so engine/coolant temperatures shown on the dash is between 80 and 40 C, it starts fine, but as soon as the accelerator is depressed when the car is in gear, it has no power; it shudders, and feels like it's about to stall out (and it has stalled twice now). When the accelerator is released, it returns to idle, which feels just a little rough. Here's the weird part, and why I don't think it's vapor lock: if it is not put in gear, I can rev the engine as high as I wish, with no problem, and doing so does not help with the stalling issue. That is, I can start it, put it in gear, and press the accelerator, producing the near stall; take it out of gear (put it in neutral or park), rev the engine quickly and smoothly; put it back in gear, press the accelerator, and it nearly stalls again. The only way to get it going is to wait--a few seconds, if the temperature has not fallen far, or up to nearly a minute if the temperature is lower. After that, though, it runs fine.
What I've tried:
I replaced the fuel pumps (which were in very bad shape), accumulator, in-line filter, and in-tank strainer. I've checked for any obvious vacuum leaks--meaning broken or disconnected lines in the engine compartment. Replaced distributor cap, rotor, ignition wires, and plugs. I've also checked for water in the new distributor cap a couple of times, and found none.
At this point, I am afraid that it might be a transmission issue.
Anybody have any insights?

Sounds to me like you may have a vacuum leak. What does the ECONOMY needle do when the car is idling and in gear? It should be pegged to the left when idling, and when you are stopped in gear. When you accelerate (whether moving or stationary) it should move to the right.

I'd baseline it by doing a smoke test, to eliminate vacuum leaks as an issue. Should be 15 minutes of a mechanic's time who has a smoke machine. Do this and report back.

test your e.h.a.

Try replacing your coil, they tend to fail when they go bad and will exhibit this kind of behavior.

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With best regards

Al