Auxiliary air valve is supposed to have a constant voltage when the engine is running! It shares the + feed with the fuel pump. It should close completely though 5 minutes after being powered up when temperature is ~20C. When the valve is not powered it should be half open when temp is ~20C, fully open when temp is -30C and completely closed when temp is +70C.
Like others said above, this sounds more like a cold start circuit problem. It's also easy to check if this is the problem. When the engine is running rough take out the power connector of the cold start injector. If problem goes away, your thermo time switch is faulty. If it doesn't go away the injector it self is leaking or the problem is somewhere else.
Cold start circuit is luckily a separate system from the ECU.

Thermo time switch controls the cold start injection directly so like said it's easy to fix.
Volvo part number for the thermo time switch is 1219731. I hope it is still available! Many parts are NLA (no longer available) but it's a Bosch part so maybe they still have those. Same switch is used in many other cars also, like 200-series Volvos with K-Jetronic injection. Also many other vehicles which have either K- or L/LE/LU-jetronic have had the same switch.
About the coolant temp sensor, there are actually two of them in injection models. One for gauge (only 1 wire going) and one for Le-Jet (sensor has two wires). Neither of those have any control over the cold start circuit because like said above the cold start circuit is a separate circuit. Faulty LE-Jet temp sensor can cause bad running also though if the ECU gets wrong info about the temperature of the engine. Condition of the ECU's temp sensor is also easy to check, the resistance between the two pins on the sensor should be about 2 or 3 kilo ohms when temperature is ~20C.
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