Heater shuts off

Is the blower motor turning off, is there a loss of heat? Does it turn back on? Is the engine coolant full?

That sounds like a bad controller. A dealer or well equipped shop can scan the climate control system and see if it is shutting off because of a bad input or if the controller itself has errors.

Does it shut off if you don't put it in gear and just set there in park?

OK if it works OK while setting still but shuts down when you put it in gear it may not be a bad control head (that is what the in dash controls are called) If it was then it should shut off irregardless of gear or motion.
I would start looking for a loose connection under the dash or at the power feeds.
GENTLY move the harness at the connections first and see if you can get it to shut down.

Different tech.

When the heater shuts off did anyone check for power to the blower motor to see if it has lost power source.

Roy

With the entire system shutting off including the power seats it has to be a power or ground loss that feeds the control head. Removing it and it still shutting off only while moving is going to make it difficult to find unless you trace the harness.

You have a loose connection on either the power or ground side. I would double check the engine grounds and then start moving the wiring. It has to be something shifting more than likely in the engine bay as that would be the only item that really moves during those times. It's making a connection when stopped, then when the engine and trans move as you start down the road it loses the connection. If this was a different model I'd say to check the vacuum lines but yours is all electronic.

With that behavior it should be setting a code. I would have it scanned to see if it's a misfire (likely) or if it is losing power to the computer. It's possible that this is related to the heater issue as both seem to happen when you apply power and the engine moves.

What is the exact code please?

If it's sending a bad signal it could be that the substituted values the ECM puts in are not good enough for wide open throttle and you are getting a lean misfire because of that. You would need to watch the live misfire data to be sure. If the code indicates a heater circuit issue and not a signal issue then you may be chasing the same issue that is causing the heater to shut off when it loses power. The only way to know would be to watch the live data from the sensors and for the voltage in the harness as it acts up.