Trying to run with the IAC valve disconnected, who knows. I never did that. I had one die, electrically open, and the car would shut off if I let it idle.
THE CTS can cause all kinds of crazy drive ability problems, idle all over the place, shut down / stall, etc. It can be bad in a way that causes this, and NOT throw a code. Reason is that it is telling the ECU how cold / hot the engine is, and this information is used to adjust how much fuel is sent in. If the mixture is way off from what it needs to be as determined by the temp, the engine won't run well, if at all.
If you have an ohmmeter, you can check it. Run the car until it's at normal temperature. Shut down. Disconnect the CTS - the 2 wire sensor on the thermostat housing. Measure the resistance. Note it. Really helpful if you have a thermometer also, so you know the temperature of the housing. Wait a while, until it cools a noticeable amount, repeat the readings. Do this a few times while it cools to ambient. #1 thing to watch for is the reading is STEADY vs jumping around. Also, it should drift in the same direction each reading.
There are temperature & resistance range notes in the FSM.
I bought an OEM CTS a few years ago, from a dealer. It was around $90.00. But the previous one lasted nearly 30 years. Not to bad, $3.00 per year.