The HVAC system is controlled by engine vacuum. When you sit at an idle, the engine intake vacuum is at its highest. When you press the accelerator, the vacuum drops.
What this is leading me to believe is you have a vacuum leak. If there is a leak, there may not be enough vacuum to keep the blend air door open on that side of the vehicle. Thus, you get heat.
If you look at pic 1 below, it shows the rear of the heater controller on the dash. I highlighted where there is a vacuum harness. First, start the engine. Turn everything off and see if you hear a hissing sound or anything that may indicate a vacuum leak.
In pic 2, I highlighted the vacuum reservoir and a few hoses. Check to make sure they are all connected, not loose, and they are made from thin plastic. Make sure they aren't cracked or broken.
Also, if you look up under the dash (passenger side I believe), you will see where one of the plastic vacuum hoses goes through the firewall into the engine compartment. Follow it to the engine and inspect it for damage as well.
One last question. Based on what is happening, I am assuming this has a manual temperature control. If it is an automatic control HVAC, let me know.
Here is a link you may find helpful. It explains how to test for an engine vacuum leak. We may be looking under the hood as well.
Images (Click to make bigger)
Saturday, May 29th, 2021 AT 8:18 PM