Here is a maybe, based on my having had problems just like this.
There is a solenoid/switch that screws into the engine block that senses the temperature of the engine/coolant and adjusts the timing through the computer to enrich the fuel mixture ( an electronic "choke").
If and when this sensor goes bad, you will get rough running when cold, as the sensor is not telling the computer the proper engine temperature.
I have had to replace 3 of these on the cars I service in the last 6 years.
God bless your efforts.
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