I say Idle Control Unit, Fuel Pump Relay or EZl is suspect. I had the same problems and on one car the ICU was bad and on another it was the EZL. The linkages can be a problem, but more often it's electrical. and the shut off and rolling to a stop is what my SEC did when the EZL went south. left to get cold it would start and run fine for an unknown amount of time and just die. A good used FPR fixed the other car.
Sounds like you may have two problems, one ICU(idle) and the EZL (dying).

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1986 Euro 500SEC "RUF" 9:1CR, "Rose"
1985 Euro 500SEC Cabriolet AMG Widebody
1982 Euro 500SEL "Blue"
2001 Texas Heeler (Aussie/Queensland X) "Sulphie",
2012 Queensland Red Heeler "Squeak"
Best dogs I've ever had.