I've had 2 slow to shutdown or no shutdown at all issues recently.
One was a torn 4 way X shaped rubber vacuum connector near the oil filter housing that has Brown, A bright yellow one way valve canister thing (Yellow goes to drivers door, Yellow with grey stripe goes to vacuum reservoir), and green with a yellow stripe coming out of it, and a grey hose feeding into it.

Brown on that connector is your shutdown "circuit", The yellow one way valve thing is central locking, green with yellow stripe is HVAC controls.

The 2nd slow shutdown I had was the floor vent pod, which is fed off that same 4 way X via the green and yellow hose that controls HVAC (goes through the HVAC valves and comes out as red with yellow stripe, should be the valve 2nd from left behind the hvac buttons).

The simple test is cap off those 2 "circuits", use a golf tee or such in the 4 way connector to leave just the brown connected, or use a mityvac on the brown hose to see if the shutdown circuit holds vacuum.

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1982 300D (w123, "Grey Car")
1982 300D (w123, "Blue Car")
2001 Ford F150 "Clifford" (The Big Red Truck)
1997 Dodge Ram 2500 12V Cummins
1996 Dodge Ram 2500 12V Cummins
Previous Vehicles:
1995 E300D, 1980 300SD, 1992 Buick Century, 2005 Saturn Ion