To go on further, the only way for idle speed to change is the addition of air, same as applying throttle by foot.
If the IAC were being pulsed more open by the PCM, there has to be a reason it is being done.
Some reasons could be the CHT varying in resistance,
The thermostat wandering in temperature responce and being detected by the CHT as large changes in coolant temp,
The IAT resistance changing in value,
A varying with temperature, air leak.
.
What sets idle on cold and hot starts is the values of the CHT and IAT inputs to the PCM that in turn sets the fuel tables, IAC opening and Timeing advance.
.
If something is changing 'out of control' with temperature you can get odd results and still may not get a code until some function goes out of limits long enough to set a code
IMO this now calls for a good tech with a high end scanner that can look at all these functions on a live data basis to catch it and see what is going out of spec or varying that should be quite stable.
Shutting off the motor enables cooling that changes the cause such that the cycle begins all over again or a reset occurs from shutdown.
An owner can only go so far in some instances without getting lucky and spending on parts not needed.
You need to some how search for it while it's happening.
The system designers gave this ability if one will go to that level of diagnostics and investigate to find it.
Even finding someone who can/will do it is a job sometimes and then it will still cost time at the labor rate plus the repair.
Good luck.

 

BINGO BANGO BONGO!!!!! I FREAKIN FOUND THE PROBLEM!!!! I decided to try a long shot and check the voltage at the battery. I did this because while running, I turned off the A/C and unplugged the fans just for giggles to see if it made a difference, and it did. With everything off and unplugged, after about 3 or 4 minutes, the idle dropped back down to normal range. From here I checked voltages. With the new fans unplugged, nothing at all else on or anything, I was getting only 13.36 volts at the battery with the truck running. I plugged in the fans and it dropped to 12.82 volts, then, turned on the A/C and headlights and it dropped to 12.22 volts. I took the alternator in to Advance Auto Parts by my house to have it tested, and their machine said it was good. The manager there told me he didn't trust the machine and said it could be a diode issue in the alternator. He gave me a discount on a brand new alternator and told me that if it didn't work, that I could bring it back in today or tomorrow and he would give me a refund and my old alternator back. I installed the new alternator, and got 14.21 volts while running with nothing on. Plugged in the fans, and it dropped to 13.19 volts. Turned on the A/C and headlights also and got 12.86 volts. I let the truck idle for about 15 minutes, which was way long enough for the problem to come back, and it idled steady between 698 and 712 RPM. I'll post back over the next few days and keep everyone informed of my findings. Bluegrass, thanks so much for giving me some ideas and direction to keep me from absolutely destroying my sanity and truck

 

I don't have a feel for what you did with the fans but if wired to come on with ignition, they take too much current running all the time without a controller and temperature sensor to run them in a reasonable manner.
Certainly the IAC and many other parts are run off the power system through fuses of course.
If the IAC has low voltage on it, the PCM control of the opening will not be correct in relation to the crank sensor rpm indictions.
The fuel injectors will be off as well as other items.
It almost sounds like the 'poor' PCM was still trying it's best to adjust to this situation by slowly returning the idle to meet table specs.
With fans and other power hungry functions you need a higher capacity battery as well as possibly a higher output alternator.
Get a Haynes manuel and look on pages 12-19 and 12-21 for the feeds and all the items powered that could be affected by low voltge.
Find the IAC and follow to all the other points that are common.
It's sometimes tough to re-engineer when your not an engineer.
Good luck.