Daniel, Vanessas car is completely different from yours, but if its the FIAV, you can fix it by replacing the trottle body.
Vanessa, the IAC is a seperate unit and you dont have to change the whole throttle body.
Daniel, if you have access to a datalogger, hook it up and see what the coolant temp sensor is telling you. Normally they fail to the cold side(start ok when cold, but shitty when hot) but it can happen the other way. If its telling the ECU that the coolant is hot, its not gonna put any extra fuel in when cold. And this is NOT the same as looking at your temp gauge, they have seperate sensors, you need a logger or other scan tool.
The other thing is, maybe you need to reset your base idle, base timing and idle stop switch. This can happen when you mod it, and can cause this issue too.
Personally, I would block off the fiav and be done with it
Just use the ISC, it works fine.