Although you two may be right about the IAC, the fact that idle increases and decreases before the engine turns off tells me the IAC is working, but that something else is giving the computer bad data, like a poor or loose, variable resistance ground on a sensor or on the entire engine, or a funky air or coolant temperature sensor, or problems with the TPS or its ground. Renix engines are notorious for variable idle speeds being caused by poor ground connections. Not just one ground wire, but all of them!
Does it die if you hold the throttle down to say 1500 rpm while it warms up? Does it die after start up only when it is cold, hot or all the time?